Properties

Label 2-936-936.277-c1-0-70
Degree 22
Conductor 936936
Sign 0.4340.900i0.434 - 0.900i
Analytic cond. 7.473997.47399
Root an. cond. 2.733862.73386
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.32 + 0.487i)2-s + (1.69 + 0.373i)3-s + (1.52 − 1.29i)4-s + (1.73 + 3.00i)5-s + (−2.42 + 0.328i)6-s − 2.47i·7-s + (−1.39 + 2.46i)8-s + (2.72 + 1.26i)9-s + (−3.76 − 3.14i)10-s + (−1.61 − 2.79i)11-s + (3.06 − 1.61i)12-s + (−3.34 − 1.35i)13-s + (1.20 + 3.29i)14-s + (1.81 + 5.72i)15-s + (0.653 − 3.94i)16-s + (2.89 + 5.01i)17-s + ⋯
L(s)  = 1  + (−0.938 + 0.344i)2-s + (0.976 + 0.215i)3-s + (0.762 − 0.646i)4-s + (0.775 + 1.34i)5-s + (−0.990 + 0.134i)6-s − 0.936i·7-s + (−0.493 + 0.869i)8-s + (0.907 + 0.420i)9-s + (−1.19 − 0.993i)10-s + (−0.487 − 0.843i)11-s + (0.884 − 0.467i)12-s + (−0.926 − 0.375i)13-s + (0.322 + 0.879i)14-s + (0.467 + 1.47i)15-s + (0.163 − 0.986i)16-s + (0.701 + 1.21i)17-s + ⋯

Functional equation

Λ(s)=(936s/2ΓC(s)L(s)=((0.4340.900i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.434 - 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(936s/2ΓC(s+1/2)L(s)=((0.4340.900i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.434 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 936936    =    2332132^{3} \cdot 3^{2} \cdot 13
Sign: 0.4340.900i0.434 - 0.900i
Analytic conductor: 7.473997.47399
Root analytic conductor: 2.733862.73386
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ936(277,)\chi_{936} (277, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 936, ( :1/2), 0.4340.900i)(2,\ 936,\ (\ :1/2),\ 0.434 - 0.900i)

Particular Values

L(1)L(1) \approx 1.43133+0.898666i1.43133 + 0.898666i
L(12)L(\frac12) \approx 1.43133+0.898666i1.43133 + 0.898666i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(1.320.487i)T 1 + (1.32 - 0.487i)T
3 1+(1.690.373i)T 1 + (-1.69 - 0.373i)T
13 1+(3.34+1.35i)T 1 + (3.34 + 1.35i)T
good5 1+(1.733.00i)T+(2.5+4.33i)T2 1 + (-1.73 - 3.00i)T + (-2.5 + 4.33i)T^{2}
7 1+2.47iT7T2 1 + 2.47iT - 7T^{2}
11 1+(1.61+2.79i)T+(5.5+9.52i)T2 1 + (1.61 + 2.79i)T + (-5.5 + 9.52i)T^{2}
17 1+(2.895.01i)T+(8.5+14.7i)T2 1 + (-2.89 - 5.01i)T + (-8.5 + 14.7i)T^{2}
19 1+(2.263.92i)T+(9.5+16.4i)T2 1 + (-2.26 - 3.92i)T + (-9.5 + 16.4i)T^{2}
23 19.31T+23T2 1 - 9.31T + 23T^{2}
29 1+(3.401.96i)T+(14.525.1i)T2 1 + (3.40 - 1.96i)T + (14.5 - 25.1i)T^{2}
31 1+(5.59+3.23i)T+(15.526.8i)T2 1 + (-5.59 + 3.23i)T + (15.5 - 26.8i)T^{2}
37 1+(1.662.87i)T+(18.532.0i)T2 1 + (1.66 - 2.87i)T + (-18.5 - 32.0i)T^{2}
41 12.56iT41T2 1 - 2.56iT - 41T^{2}
43 19.40iT43T2 1 - 9.40iT - 43T^{2}
47 1+(7.93+4.57i)T+(23.5+40.7i)T2 1 + (7.93 + 4.57i)T + (23.5 + 40.7i)T^{2}
53 10.880iT53T2 1 - 0.880iT - 53T^{2}
59 1+(4.67+8.09i)T+(29.551.0i)T2 1 + (-4.67 + 8.09i)T + (-29.5 - 51.0i)T^{2}
61 1+12.6iT61T2 1 + 12.6iT - 61T^{2}
67 1+3.65T+67T2 1 + 3.65T + 67T^{2}
71 1+(3.311.91i)T+(35.561.4i)T2 1 + (3.31 - 1.91i)T + (35.5 - 61.4i)T^{2}
73 1+6.92iT73T2 1 + 6.92iT - 73T^{2}
79 1+(5.769.99i)T+(39.568.4i)T2 1 + (5.76 - 9.99i)T + (-39.5 - 68.4i)T^{2}
83 1+(1.071.86i)T+(41.571.8i)T2 1 + (1.07 - 1.86i)T + (-41.5 - 71.8i)T^{2}
89 1+(1.67+0.964i)T+(44.5+77.0i)T2 1 + (1.67 + 0.964i)T + (44.5 + 77.0i)T^{2}
97 1+5.53iT97T2 1 + 5.53iT - 97T^{2}
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   L(s)=p j=12(1αj,pps)1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−9.964704638453795236912296838263, −9.704329892686110152046446505937, −8.378465636182048336616705988597, −7.75691997805107698673856682838, −7.06493655858251159553227894951, −6.24885429073123646576956806240, −5.15264707265935279447055567600, −3.37957626673582434836735811855, −2.79599798944543426730691111469, −1.46478118585463904424441606733, 1.08629804717760564042892262567, 2.26920103125289776549597752972, 2.88232555875016317050562788605, 4.67209280832052748577046586245, 5.40939807609641499522197616645, 7.01040290588439635361953595146, 7.48077040965802014854332265496, 8.681586515020954770600510034464, 9.088987661161376868761418268954, 9.550286548385151835811706696527

Graph of the ZZ-function along the critical line