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Properties

Label 2-351-117.88-c3-0-34
Degree $2$
Conductor $351$
Sign $-0.717 - 0.696i$
Analytic cond. $20.7096$
Root an. cond. $4.55078$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.67i·2-s − 5.47·4-s + (−15.6 + 9.05i)5-s + (27.0 − 15.5i)7-s − 9.27i·8-s + (33.2 + 57.5i)10-s + 5.30i·11-s + (−3.93 − 46.7i)13-s + (−57.2 − 99.1i)14-s − 77.8·16-s + (1.95 − 3.38i)17-s + (19.5 + 11.3i)19-s + (85.8 − 49.5i)20-s + 19.4·22-s + (27.9 − 48.4i)23-s + ⋯
L(s)  = 1  − 1.29i·2-s − 0.684·4-s + (−1.40 + 0.809i)5-s + (1.45 − 0.842i)7-s − 0.409i·8-s + (1.05 + 1.82i)10-s + 0.145i·11-s + (−0.0839 − 0.996i)13-s + (−1.09 − 1.89i)14-s − 1.21·16-s + (0.0279 − 0.0483i)17-s + (0.236 + 0.136i)19-s + (0.959 − 0.554i)20-s + 0.188·22-s + (0.253 − 0.439i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.717 - 0.696i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.717 - 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(351\)    =    \(3^{3} \cdot 13\)
Sign: $-0.717 - 0.696i$
Analytic conductor: \(20.7096\)
Root analytic conductor: \(4.55078\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{351} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 351,\ (\ :3/2),\ -0.717 - 0.696i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8111783225\)
\(L(\frac12)\) \(\approx\) \(0.8111783225\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 + (3.93 + 46.7i)T \)
good2 \( 1 + 3.67iT - 8T^{2} \)
5 \( 1 + (15.6 - 9.05i)T + (62.5 - 108. i)T^{2} \)
7 \( 1 + (-27.0 + 15.5i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 - 5.30iT - 1.33e3T^{2} \)
17 \( 1 + (-1.95 + 3.38i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-19.5 - 11.3i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-27.9 + 48.4i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 294.T + 2.43e4T^{2} \)
31 \( 1 + (150. - 86.9i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (328. - 189. i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (114. + 66.3i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-30.3 - 52.4i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (229. + 132. i)T + (5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 446.T + 1.48e5T^{2} \)
59 \( 1 + 253. iT - 2.05e5T^{2} \)
61 \( 1 + (176. + 304. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (387. + 223. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-260. - 150. i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + 86.9iT - 3.89e5T^{2} \)
79 \( 1 + (243. - 422. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (842. + 486. i)T + (2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-789. + 456. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (11.9 - 6.87i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−10.81128653748722689021186053578, −10.11912051298909644082333184934, −8.572457230935937522277372925177, −7.60567434695593794875303966412, −7.05333569804087618068387435630, −5.03851980261875483006798085077, −3.93310123316932207067208864230, −3.23829449592587807328133560290, −1.70335101540898390167107361904, −0.28289551014870310835627348196, 1.82157317700285179728408274787, 3.99926471755471924110490707208, 4.97891447916612512760086354420, 5.64266451004929865793407222176, 7.24640685357554540452367382525, 7.70755261355888500325821071252, 8.690140016725698624054779290788, 9.025138157363749420083624907522, 11.21846034669457134055921572733, 11.50452663387179677998217079269

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