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Properties

Label 2-354-177.176-c5-0-14
Degree $2$
Conductor $354$
Sign $-0.596 - 0.802i$
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 + (14.0 − 6.82i)3-s + 16·4-s + 34.7i·5-s + (56.0 − 27.2i)6-s − 97.2·7-s + 64·8-s + (149. − 191. i)9-s + 139. i·10-s − 627.·11-s + (224. − 109. i)12-s + 675. i·13-s − 389.·14-s + (237. + 487. i)15-s + 256·16-s + 560. i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.899 − 0.437i)3-s + 0.5·4-s + 0.622i·5-s + (0.635 − 0.309i)6-s − 0.750·7-s + 0.353·8-s + (0.616 − 0.787i)9-s + 0.439i·10-s − 1.56·11-s + (0.449 − 0.218i)12-s + 1.10i·13-s − 0.530·14-s + (0.272 + 0.559i)15-s + 0.250·16-s + 0.470i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.596 - 0.802i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.596 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-0.596 - 0.802i$
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: \((2,\ 354,\ (\ :5/2),\ -0.596 - 0.802i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.796452298\)
\(L(\frac12)\) \(\approx\) \(1.796452298\)
\(L(\frac{7}{2})\) 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 \( 1 + (-14.0 + 6.82i)T \)
59 \( 1 + (4.95e3 + 2.62e4i)T \)
good5 \( 1 - 34.7iT - 3.12e3T^{2} \)
7 \( 1 + 97.2T + 1.68e4T^{2} \)
11 \( 1 + 627.T + 1.61e5T^{2} \)
13 \( 1 - 675. iT - 3.71e5T^{2} \)
17 \( 1 - 560. iT - 1.41e6T^{2} \)
19 \( 1 + 2.62e3T + 2.47e6T^{2} \)
23 \( 1 + 334.T + 6.43e6T^{2} \)
29 \( 1 - 5.74e3iT - 2.05e7T^{2} \)
31 \( 1 - 8.35e3iT - 2.86e7T^{2} \)
37 \( 1 - 6.56e3iT - 6.93e7T^{2} \)
41 \( 1 + 7.86e3iT - 1.15e8T^{2} \)
43 \( 1 + 1.29e4iT - 1.47e8T^{2} \)
47 \( 1 + 2.12e4T + 2.29e8T^{2} \)
53 \( 1 - 4.00e3iT - 4.18e8T^{2} \)
61 \( 1 - 2.69e4iT - 8.44e8T^{2} \)
67 \( 1 + 1.55e4iT - 1.35e9T^{2} \)
71 \( 1 + 2.55e4iT - 1.80e9T^{2} \)
73 \( 1 - 4.51e4iT - 2.07e9T^{2} \)
79 \( 1 - 9.64e4T + 3.07e9T^{2} \)
83 \( 1 - 4.96e4T + 3.93e9T^{2} \)
89 \( 1 + 2.01e4T + 5.58e9T^{2} \)
97 \( 1 - 4.14e4iT - 8.58e9T^{2} \)
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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

−10.80543204804180742790718693136, −10.27497742879572989294163319159, −8.973844236510888362349076203585, −8.087910063776111734954962744788, −6.85378801841904825035788595187, −6.53116285113289175572877463193, −4.95432424805132362004621075384, −3.65624391251943017220150717200, −2.80533123656551942351247893347, −1.85210644165008936856353043780, 0.27125587828653873189755480444, 2.29178167809796857772054521383, 3.05332887472101186098837213599, 4.28955142844213175078337414625, 5.15362991077495408485021163642, 6.26538574482696942378571286977, 7.75264407642380371432604738297, 8.238188325241208114830897462101, 9.517267812048557235195038169764, 10.26351906384304311011764625234

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