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Properties

Label 2-700-5.4-c5-0-21
Degree 22
Conductor 700700
Sign 0.894+0.447i0.894 + 0.447i
Analytic cond. 112.268112.268
Root an. cond. 10.595610.5956
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·3-s + 49i·7-s + 239·9-s − 720·11-s − 572i·13-s + 1.25e3i·17-s + 94·19-s − 98·21-s − 96i·23-s + 964i·27-s + 4.37e3·29-s − 6.24e3·31-s − 1.44e3i·33-s − 1.07e4i·37-s + 1.14e3·39-s + ⋯
L(s)  = 1  + 0.128i·3-s + 0.377i·7-s + 0.983·9-s − 1.79·11-s − 0.938i·13-s + 1.05i·17-s + 0.0597·19-s − 0.0484·21-s − 0.0378i·23-s + 0.254i·27-s + 0.965·29-s − 1.16·31-s − 0.230i·33-s − 1.29i·37-s + 0.120·39-s + ⋯

Functional equation

Λ(s)=(700s/2 ΓC(s) L(s)=((0.894+0.447i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(700s/2 ΓC(s+5/2) L(s)=((0.894+0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 700700    =    22⋅52⋅72^{2} \cdot 5^{2} \cdot 7
Sign: 0.894+0.447i0.894 + 0.447i
Analytic conductor: 112.268112.268
Root analytic conductor: 10.595610.5956
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ700(449,⋅)\chi_{700} (449, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 700, ( :5/2), 0.894+0.447i)(2,\ 700,\ (\ :5/2),\ 0.894 + 0.447i)

Particular Values

L(3)L(3) ≈\approx 1.7519066711.751906671
L(12)L(\frac12) ≈\approx 1.7519066711.751906671
L(72)L(\frac{7}{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
5 1 1
7 1−49iT 1 - 49iT
good3 1−2iT−243T2 1 - 2iT - 243T^{2}
11 1+720T+1.61e5T2 1 + 720T + 1.61e5T^{2}
13 1+572iT−3.71e5T2 1 + 572iT - 3.71e5T^{2}
17 1−1.25e3iT−1.41e6T2 1 - 1.25e3iT - 1.41e6T^{2}
19 1−94T+2.47e6T2 1 - 94T + 2.47e6T^{2}
23 1+96iT−6.43e6T2 1 + 96iT - 6.43e6T^{2}
29 1−4.37e3T+2.05e7T2 1 - 4.37e3T + 2.05e7T^{2}
31 1+6.24e3T+2.86e7T2 1 + 6.24e3T + 2.86e7T^{2}
37 1+1.07e4iT−6.93e7T2 1 + 1.07e4iT - 6.93e7T^{2}
41 1−1.20e4T+1.15e8T2 1 - 1.20e4T + 1.15e8T^{2}
43 1−9.16e3iT−1.47e8T2 1 - 9.16e3iT - 1.47e8T^{2}
47 1+2.58e4iT−2.29e8T2 1 + 2.58e4iT - 2.29e8T^{2}
53 1+1.01e3iT−4.18e8T2 1 + 1.01e3iT - 4.18e8T^{2}
59 1+1.24e3T+7.14e8T2 1 + 1.24e3T + 7.14e8T^{2}
61 1−7.59e3T+8.44e8T2 1 - 7.59e3T + 8.44e8T^{2}
67 1−4.11e4iT−1.35e9T2 1 - 4.11e4iT - 1.35e9T^{2}
71 1+3.76e4T+1.80e9T2 1 + 3.76e4T + 1.80e9T^{2}
73 1−1.34e4iT−2.07e9T2 1 - 1.34e4iT - 2.07e9T^{2}
79 1+6.24e3T+3.07e9T2 1 + 6.24e3T + 3.07e9T^{2}
83 1−2.52e4iT−3.93e9T2 1 - 2.52e4iT - 3.93e9T^{2}
89 1−4.51e4T+5.58e9T2 1 - 4.51e4T + 5.58e9T^{2}
97 1−1.07e5iT−8.58e9T2 1 - 1.07e5iT - 8.58e9T^{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

−9.832570293895776412158818852839, −8.649360719332333577174406183918, −7.88655519999573719184279539080, −7.16343352796214164402195215981, −5.85450451788696116084132978317, −5.21057904183596230082472839860, −4.11241372451629510421263495628, −2.95206904774621332742341164589, −1.92846793092497751727320626817, −0.49616688747952715104488793260, 0.75869988972999107093360258247, 2.02340381905979202935592871561, 3.08161184393776156385168972222, 4.42484505528440347936946747270, 5.05436040062549233605444697164, 6.30834623855936092748459620186, 7.32822126539191534170519903721, 7.73690482657648142927380389997, 8.958250196990659294690422451671, 9.848457870421664957455084023860

Graph of the ZZ-function along the critical line