Properties

Label 2-35-7.2-c5-0-8
Degree $2$
Conductor $35$
Sign $0.652 + 0.757i$
Analytic cond. $5.61343$
Root an. cond. $2.36926$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.398 − 0.689i)2-s + (−5.28 − 9.15i)3-s + (15.6 + 27.1i)4-s + (12.5 − 21.6i)5-s − 8.42·6-s + (103. − 78.1i)7-s + 50.4·8-s + (65.5 − 113. i)9-s + (−9.95 − 17.2i)10-s + (−262. − 454. i)11-s + (165. − 287. i)12-s + 819.·13-s + (−12.7 − 102. i)14-s − 264.·15-s + (−481. + 834. i)16-s + (460. + 797. i)17-s + ⋯
L(s)  = 1  + (0.0704 − 0.121i)2-s + (−0.339 − 0.587i)3-s + (0.490 + 0.848i)4-s + (0.223 − 0.387i)5-s − 0.0955·6-s + (0.797 − 0.602i)7-s + 0.278·8-s + (0.269 − 0.467i)9-s + (−0.0314 − 0.0545i)10-s + (−0.654 − 1.13i)11-s + (0.332 − 0.575i)12-s + 1.34·13-s + (−0.0173 − 0.139i)14-s − 0.303·15-s + (−0.470 + 0.814i)16-s + (0.386 + 0.669i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.652 + 0.757i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.652 + 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.652 + 0.757i$
Analytic conductor: \(5.61343\)
Root analytic conductor: \(2.36926\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :5/2),\ 0.652 + 0.757i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.63044 - 0.747978i\)
\(L(\frac12)\) \(\approx\) \(1.63044 - 0.747978i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-12.5 + 21.6i)T \)
7 \( 1 + (-103. + 78.1i)T \)
good2 \( 1 + (-0.398 + 0.689i)T + (-16 - 27.7i)T^{2} \)
3 \( 1 + (5.28 + 9.15i)T + (-121.5 + 210. i)T^{2} \)
11 \( 1 + (262. + 454. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 - 819.T + 3.71e5T^{2} \)
17 \( 1 + (-460. - 797. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-348. + 602. i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (1.19e3 - 2.06e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + 6.20e3T + 2.05e7T^{2} \)
31 \( 1 + (-4.00e3 - 6.93e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-4.06e3 + 7.04e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 - 1.70e3T + 1.15e8T^{2} \)
43 \( 1 + 8.13e3T + 1.47e8T^{2} \)
47 \( 1 + (-1.66e3 + 2.88e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-1.63e4 - 2.83e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (1.61e4 + 2.80e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (2.32e4 - 4.02e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-2.67e3 - 4.63e3i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 4.44e4T + 1.80e9T^{2} \)
73 \( 1 + (-3.52e4 - 6.10e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (2.31e4 - 4.01e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 2.72e4T + 3.93e9T^{2} \)
89 \( 1 + (-3.23e4 + 5.59e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 3.14e4T + 8.58e9T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.61194919567368715409121877073, −13.72729542458508922243512072455, −12.96445151021362257425511480809, −11.68190273778064935621004578665, −10.77730824146888241073518609430, −8.580728850122729915259954914506, −7.48740189969178515130448262714, −5.92865914559039565499175120632, −3.70777367665338767091578991471, −1.30766329564277988563762213469, 1.98092726636832571472988568898, 4.78956070308716691732174086983, 5.96094088324118049966136020493, 7.71194181955710678317283342843, 9.722779087087750668048483109683, 10.67592391577261926139050047503, 11.63945744744337492255453913633, 13.49769518184715194966211613369, 14.83739050792373139940599890299, 15.51986083622114552024401465797

Graph of the $Z$-function along the critical line