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Properties

Label 2-354-177.176-c5-0-24
Degree $2$
Conductor $354$
Sign $0.727 + 0.686i$
Analytic cond. $56.7758$
Root an. cond. $7.53497$
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  − 4·2-s + (−11.5 − 10.4i)3-s + 16·4-s − 67.9i·5-s + (46.0 + 41.9i)6-s − 61.6·7-s − 64·8-s + (22.5 + 241. i)9-s + 271. i·10-s − 640.·11-s + (−184. − 167. i)12-s − 578. i·13-s + 246.·14-s + (−713. + 783. i)15-s + 256·16-s + 1.37e3i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.739 − 0.673i)3-s + 0.5·4-s − 1.21i·5-s + (0.522 + 0.476i)6-s − 0.475·7-s − 0.353·8-s + (0.0928 + 0.995i)9-s + 0.859i·10-s − 1.59·11-s + (−0.369 − 0.336i)12-s − 0.948i·13-s + 0.336·14-s + (−0.819 + 0.899i)15-s + 0.250·16-s + 1.15i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.686i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.727 + 0.686i$
Analytic conductor: \(56.7758\)
Root analytic conductor: \(7.53497\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :5/2),\ 0.727 + 0.686i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.6471333357\)
\(L(\frac12)\) \(\approx\) \(0.6471333357\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 4T \)
3 \( 1 + (11.5 + 10.4i)T \)
59 \( 1 + (-2.67e4 - 479. i)T \)
good5 \( 1 + 67.9iT - 3.12e3T^{2} \)
7 \( 1 + 61.6T + 1.68e4T^{2} \)
11 \( 1 + 640.T + 1.61e5T^{2} \)
13 \( 1 + 578. iT - 3.71e5T^{2} \)
17 \( 1 - 1.37e3iT - 1.41e6T^{2} \)
19 \( 1 - 2.52e3T + 2.47e6T^{2} \)
23 \( 1 + 625.T + 6.43e6T^{2} \)
29 \( 1 - 5.93e3iT - 2.05e7T^{2} \)
31 \( 1 + 491. iT - 2.86e7T^{2} \)
37 \( 1 - 1.35e4iT - 6.93e7T^{2} \)
41 \( 1 - 780. iT - 1.15e8T^{2} \)
43 \( 1 + 4.53e3iT - 1.47e8T^{2} \)
47 \( 1 + 1.83e4T + 2.29e8T^{2} \)
53 \( 1 + 1.72e4iT - 4.18e8T^{2} \)
61 \( 1 - 854. iT - 8.44e8T^{2} \)
67 \( 1 + 2.85e4iT - 1.35e9T^{2} \)
71 \( 1 - 5.29e4iT - 1.80e9T^{2} \)
73 \( 1 - 5.07e4iT - 2.07e9T^{2} \)
79 \( 1 - 9.94e4T + 3.07e9T^{2} \)
83 \( 1 - 6.49e4T + 3.93e9T^{2} \)
89 \( 1 + 1.22e5T + 5.58e9T^{2} \)
97 \( 1 + 828. iT - 8.58e9T^{2} \)
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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

−10.42469460140130991653620576310, −9.767464744764672306064005726022, −8.306532524247136743118114798355, −7.999737042893990478381347401206, −6.81850409108834535085182331134, −5.54685526509690008340740348837, −5.08240326401173969539633446467, −3.05871431704876812523267673041, −1.53788538122020876399211021041, −0.57446140045712486711780423509, 0.43881430539397872166118002740, 2.48857570448672308221210082059, 3.42014392135597676013349228707, 4.99461383060062081277337041449, 6.04903166286638380711900738155, 7.00540613694581249721995368498, 7.73742196932929232935771094679, 9.371895575518284912670473641799, 9.837414477809884630271624829401, 10.70104304407990286993479040427

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