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Properties

Label 2-1352-13.3-c1-0-35
Degree 22
Conductor 13521352
Sign −0.477+0.878i-0.477 + 0.878i
Analytic cond. 10.795710.7957
Root an. cond. 3.285693.28569
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  + (−0.5 − 0.866i)3-s + 5-s + (2.5 − 4.33i)7-s + (1 − 1.73i)9-s + (−1 − 1.73i)11-s + (−0.5 − 0.866i)15-s + (1.5 − 2.59i)17-s + (−1 + 1.73i)19-s − 5·21-s + (−2 − 3.46i)23-s − 4·25-s − 5·27-s + (3 + 5.19i)29-s + 4·31-s + (−0.999 + 1.73i)33-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + 0.447·5-s + (0.944 − 1.63i)7-s + (0.333 − 0.577i)9-s + (−0.301 − 0.522i)11-s + (−0.129 − 0.223i)15-s + (0.363 − 0.630i)17-s + (−0.229 + 0.397i)19-s − 1.09·21-s + (−0.417 − 0.722i)23-s − 0.800·25-s − 0.962·27-s + (0.557 + 0.964i)29-s + 0.718·31-s + (−0.174 + 0.301i)33-s + ⋯

Functional equation

Λ(s)=(1352s/2 ΓC(s) L(s)=((−0.477+0.878i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.477 + 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1352s/2 ΓC(s+1/2) L(s)=((−0.477+0.878i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13521352    =    23⋅1322^{3} \cdot 13^{2}
Sign: −0.477+0.878i-0.477 + 0.878i
Analytic conductor: 10.795710.7957
Root analytic conductor: 3.285693.28569
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1352(1329,⋅)\chi_{1352} (1329, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1352, ( :1/2), −0.477+0.878i)(2,\ 1352,\ (\ :1/2),\ -0.477 + 0.878i)

Particular Values

L(1)L(1) ≈\approx 1.7057652101.705765210
L(12)L(\frac12) ≈\approx 1.7057652101.705765210
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
13 1 1
good3 1+(0.5+0.866i)T+(−1.5+2.59i)T2 1 + (0.5 + 0.866i)T + (-1.5 + 2.59i)T^{2}
5 1−T+5T2 1 - T + 5T^{2}
7 1+(−2.5+4.33i)T+(−3.5−6.06i)T2 1 + (-2.5 + 4.33i)T + (-3.5 - 6.06i)T^{2}
11 1+(1+1.73i)T+(−5.5+9.52i)T2 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2}
17 1+(−1.5+2.59i)T+(−8.5−14.7i)T2 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2}
19 1+(1−1.73i)T+(−9.5−16.4i)T2 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2}
23 1+(2+3.46i)T+(−11.5+19.9i)T2 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2}
29 1+(−3−5.19i)T+(−14.5+25.1i)T2 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2}
31 1−4T+31T2 1 - 4T + 31T^{2}
37 1+(−5.5−9.52i)T+(−18.5+32.0i)T2 1 + (-5.5 - 9.52i)T + (-18.5 + 32.0i)T^{2}
41 1+(−4−6.92i)T+(−20.5+35.5i)T2 1 + (-4 - 6.92i)T + (-20.5 + 35.5i)T^{2}
43 1+(−0.5+0.866i)T+(−21.5−37.2i)T2 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2}
47 1+9T+47T2 1 + 9T + 47T^{2}
53 1+12T+53T2 1 + 12T + 53T^{2}
59 1+(−3+5.19i)T+(−29.5−51.0i)T2 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2}
61 1+(−30.5−52.8i)T2 1 + (-30.5 - 52.8i)T^{2}
67 1+(−3−5.19i)T+(−33.5+58.0i)T2 1 + (-3 - 5.19i)T + (-33.5 + 58.0i)T^{2}
71 1+(−3.5+6.06i)T+(−35.5−61.4i)T2 1 + (-3.5 + 6.06i)T + (-35.5 - 61.4i)T^{2}
73 1−2T+73T2 1 - 2T + 73T^{2}
79 1−12T+79T2 1 - 12T + 79T^{2}
83 1−16T+83T2 1 - 16T + 83T^{2}
89 1+(5+8.66i)T+(−44.5+77.0i)T2 1 + (5 + 8.66i)T + (-44.5 + 77.0i)T^{2}
97 1+(5−8.66i)T+(−48.5−84.0i)T2 1 + (5 - 8.66i)T + (-48.5 - 84.0i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−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

−9.599525352473124127169844373811, −8.121300936278992292384728203382, −7.87726789146238762715955008572, −6.75363348897079475823065785259, −6.29573227746206999082937061923, −5.03815395643918309518748986800, −4.29108359767455476537091642877, −3.20376293241790886983210112969, −1.62712338676556952675748799845, −0.75371846909473262727840231723, 1.85693700839219984660689538115, 2.44677094516806791210642899994, 4.07814090755880214648945266389, 4.98051144552532696103142242950, 5.57644483995376312148707495966, 6.28171384778976307342254895680, 7.78651675301647908062050144506, 8.108668184087682630282890321462, 9.293951780450625961603546924698, 9.718383272404965103631726780028

Graph of the ZZ-function along the critical line