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Properties

Label 2-351-117.25-c3-0-18
Degree $2$
Conductor $351$
Sign $0.672 + 0.740i$
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  + (−2.81 − 1.62i)2-s + (1.28 + 2.22i)4-s + (8.08 − 4.67i)5-s + (−5.60 − 3.23i)7-s + 17.6i·8-s − 30.3·10-s + (28.4 + 16.4i)11-s + (−28.5 − 37.2i)13-s + (10.5 + 18.2i)14-s + (38.9 − 67.4i)16-s + 99.5·17-s + 136. i·19-s + (20.7 + 11.9i)20-s + (−53.4 − 92.5i)22-s + (7.91 + 13.7i)23-s + ⋯
L(s)  = 1  + (−0.995 − 0.574i)2-s + (0.160 + 0.277i)4-s + (0.723 − 0.417i)5-s + (−0.302 − 0.174i)7-s + 0.780i·8-s − 0.960·10-s + (0.780 + 0.450i)11-s + (−0.608 − 0.793i)13-s + (0.200 + 0.347i)14-s + (0.608 − 1.05i)16-s + 1.41·17-s + 1.64i·19-s + (0.231 + 0.133i)20-s + (−0.518 − 0.897i)22-s + (0.0717 + 0.124i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.672 + 0.740i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.672 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(351\)    =    \(3^{3} \cdot 13\)
Sign: $0.672 + 0.740i$
Analytic conductor: \(20.7096\)
Root analytic conductor: \(4.55078\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{351} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 351,\ (\ :3/2),\ 0.672 + 0.740i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.184766640\)
\(L(\frac12)\) \(\approx\) \(1.184766640\)
\(L(\frac{5}{2})\) 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 \( 1 + (28.5 + 37.2i)T \)
good2 \( 1 + (2.81 + 1.62i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + (-8.08 + 4.67i)T + (62.5 - 108. i)T^{2} \)
7 \( 1 + (5.60 + 3.23i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (-28.4 - 16.4i)T + (665.5 + 1.15e3i)T^{2} \)
17 \( 1 - 99.5T + 4.91e3T^{2} \)
19 \( 1 - 136. iT - 6.85e3T^{2} \)
23 \( 1 + (-7.91 - 13.7i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (29.1 - 50.5i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-80.0 + 46.2i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 384. iT - 5.06e4T^{2} \)
41 \( 1 + (33.4 - 19.2i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-240. + 417. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-232. - 134. i)T + (5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 475.T + 1.48e5T^{2} \)
59 \( 1 + (-347. + 200. i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (119. - 207. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-671. + 387. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 769. iT - 3.57e5T^{2} \)
73 \( 1 + 501. iT - 3.89e5T^{2} \)
79 \( 1 + (-418. + 725. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (19.4 + 11.2i)T + (2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 126. iT - 7.04e5T^{2} \)
97 \( 1 + (-72.1 - 41.6i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−10.46130968474384114845344706360, −9.961927326706404705006477887109, −9.411420532199914307308930189393, −8.326729810631518710428370997611, −7.46346426435953480445591082309, −5.95914035657214168796894148657, −5.14445227640718349812244395373, −3.45756867384797789636274382124, −1.92612026235039351490361751288, −0.917765446918482312007002624074, 0.845970067580981611032827200361, 2.59441461154332360799339531240, 4.05566692926692201941232645433, 5.68292427723507198268412786004, 6.64973765977398825653784560725, 7.30806611931726842798278957787, 8.505635834535135710871356870670, 9.416527484155424073477034042727, 9.777824075928563023993483092812, 10.93522460681869478050097920823

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