Properties

Label 1-4729-4729.3282-r0-0-0
Degree 11
Conductor 47294729
Sign 0.540+0.841i-0.540 + 0.841i
Analytic cond. 21.961321.9613
Root an. cond. 21.961321.9613
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.709 − 0.704i)2-s + (−0.949 + 0.313i)3-s + (0.00797 − 0.999i)4-s + (0.140 − 0.990i)5-s + (−0.453 + 0.891i)6-s + (0.631 − 0.775i)7-s + (−0.698 − 0.715i)8-s + (0.803 − 0.595i)9-s + (−0.597 − 0.801i)10-s + (0.768 − 0.639i)11-s + (0.305 + 0.952i)12-s + (−0.875 + 0.483i)13-s + (−0.0981 − 0.995i)14-s + (0.177 + 0.984i)15-s + (−0.999 − 0.0159i)16-s + (0.242 − 0.970i)17-s + ⋯
L(s)  = 1  + (0.709 − 0.704i)2-s + (−0.949 + 0.313i)3-s + (0.00797 − 0.999i)4-s + (0.140 − 0.990i)5-s + (−0.453 + 0.891i)6-s + (0.631 − 0.775i)7-s + (−0.698 − 0.715i)8-s + (0.803 − 0.595i)9-s + (−0.597 − 0.801i)10-s + (0.768 − 0.639i)11-s + (0.305 + 0.952i)12-s + (−0.875 + 0.483i)13-s + (−0.0981 − 0.995i)14-s + (0.177 + 0.984i)15-s + (−0.999 − 0.0159i)16-s + (0.242 − 0.970i)17-s + ⋯

Functional equation

Λ(s)=(4729s/2ΓR(s)L(s)=((0.540+0.841i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.540 + 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(4729s/2ΓR(s)L(s)=((0.540+0.841i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.540 + 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 47294729
Sign: 0.540+0.841i-0.540 + 0.841i
Analytic conductor: 21.961321.9613
Root analytic conductor: 21.961321.9613
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ4729(3282,)\chi_{4729} (3282, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 4729, (0: ), 0.540+0.841i)(1,\ 4729,\ (0:\ ),\ -0.540 + 0.841i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.63547820781.164000206i-0.6354782078 - 1.164000206i
L(12)L(\frac12) \approx 0.63547820781.164000206i-0.6354782078 - 1.164000206i
L(1)L(1) \approx 0.76233113600.8398107802i0.7623311360 - 0.8398107802i
L(1)L(1) \approx 0.76233113600.8398107802i0.7623311360 - 0.8398107802i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad4729 1 1
good2 1+(0.7090.704i)T 1 + (0.709 - 0.704i)T
3 1+(0.949+0.313i)T 1 + (-0.949 + 0.313i)T
5 1+(0.1400.990i)T 1 + (0.140 - 0.990i)T
7 1+(0.6310.775i)T 1 + (0.631 - 0.775i)T
11 1+(0.7680.639i)T 1 + (0.768 - 0.639i)T
13 1+(0.875+0.483i)T 1 + (-0.875 + 0.483i)T
17 1+(0.2420.970i)T 1 + (0.242 - 0.970i)T
19 1+(0.749+0.661i)T 1 + (0.749 + 0.661i)T
23 1+(0.06370.997i)T 1 + (-0.0637 - 0.997i)T
29 1+(0.704+0.709i)T 1 + (-0.704 + 0.709i)T
31 1+(0.956+0.290i)T 1 + (0.956 + 0.290i)T
37 1+(0.1290.991i)T 1 + (-0.129 - 0.991i)T
41 1+(0.174+0.984i)T 1 + (0.174 + 0.984i)T
43 1+(0.972+0.234i)T 1 + (-0.972 + 0.234i)T
47 1+(0.881+0.472i)T 1 + (0.881 + 0.472i)T
53 1+(0.4500.892i)T 1 + (-0.450 - 0.892i)T
59 1+(0.7130.700i)T 1 + (-0.713 - 0.700i)T
61 1+(0.6850.728i)T 1 + (-0.685 - 0.728i)T
67 1+(0.647+0.762i)T 1 + (-0.647 + 0.762i)T
71 1+(0.0132+0.999i)T 1 + (0.0132 + 0.999i)T
73 1+(0.370+0.928i)T 1 + (-0.370 + 0.928i)T
79 1+(0.8660.5i)T 1 + (0.866 - 0.5i)T
83 1+(0.375+0.926i)T 1 + (-0.375 + 0.926i)T
89 1+(0.9870.156i)T 1 + (-0.987 - 0.156i)T
97 1+(0.631+0.775i)T 1 + (-0.631 + 0.775i)T
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   L(s)=p (1αpps)1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−18.28144381563708663102662888552, −17.753355904282104530428475310750, −17.187917092061963060854951787439, −16.903148543532777308805201611928, −15.49408684159581564896186522213, −15.319220709939558329674405484594, −14.82430136133392842399370258876, −13.830944313505608489922842111951, −13.42318464330508339663797481213, −12.294450083267708204520924416114, −12.00345758887371008104182378565, −11.48532606241591183272053481821, −10.64496978224513914421600314392, −9.8188744871852133399053009238, −9.0136970104477367095319385073, −7.71439972194042586168626537899, −7.60756841734879893303719492041, −6.73259795846136359445604793567, −6.07209084802445589160245270754, −5.550482528549180787045186714121, −4.84147057016961026874923498617, −4.10679121625674401296785237563, −3.11996258013803350331912114771, −2.27668833180681335880955951524, −1.52518846005755041506162427586, 0.31721934354165124204778947743, 1.17420349780905446851331756926, 1.620833931418219983385442478164, 2.92976835321432201679966906921, 3.96050601543604621438799261659, 4.393679853739126585436584474596, 5.08703818385370654333498610883, 5.51039552766609448883297201904, 6.46679449720705400379981889214, 7.10484433937828798782416713055, 8.14750720691657749986300646466, 9.25147443943877342117121439361, 9.652017211409536773095795923939, 10.36641547111430889299146858860, 11.201616409473186538982022183, 11.65323739736837849550791333682, 12.23462339820773742519988607005, 12.77126240193625273017533999045, 13.73426670892054171386230529093, 14.209252936323112123254514474784, 14.80177748312658213539164527804, 15.94243116771142902650669064465, 16.42377058231898290425587615795, 16.89730032468917199354199385877, 17.67410988547640522654713066309

Graph of the ZZ-function along the critical line