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Properties

Label 2-354-177.176-c3-0-46
Degree $2$
Conductor $354$
Sign $0.907 + 0.420i$
Analytic cond. $20.8866$
Root an. cond. $4.57019$
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  + 2·2-s + (5.16 − 0.541i)3-s + 4·4-s − 1.12i·5-s + (10.3 − 1.08i)6-s − 8.16·7-s + 8·8-s + (26.4 − 5.59i)9-s − 2.24i·10-s + 24.0·11-s + (20.6 − 2.16i)12-s − 67.9i·13-s − 16.3·14-s + (−0.608 − 5.80i)15-s + 16·16-s + 20.4i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.994 − 0.104i)3-s + 0.5·4-s − 0.100i·5-s + (0.703 − 0.0736i)6-s − 0.441·7-s + 0.353·8-s + (0.978 − 0.207i)9-s − 0.0710i·10-s + 0.658·11-s + (0.497 − 0.0521i)12-s − 1.45i·13-s − 0.311·14-s + (−0.0104 − 0.0999i)15-s + 0.250·16-s + 0.291i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 + 0.420i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.907 + 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.907 + 0.420i$
Analytic conductor: \(20.8866\)
Root analytic conductor: \(4.57019\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :3/2),\ 0.907 + 0.420i)\)

Particular Values

\(L(2)\) \(\approx\) \(4.269735498\)
\(L(\frac12)\) \(\approx\) \(4.269735498\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2T \)
3 \( 1 + (-5.16 + 0.541i)T \)
59 \( 1 + (388. + 232. i)T \)
good5 \( 1 + 1.12iT - 125T^{2} \)
7 \( 1 + 8.16T + 343T^{2} \)
11 \( 1 - 24.0T + 1.33e3T^{2} \)
13 \( 1 + 67.9iT - 2.19e3T^{2} \)
17 \( 1 - 20.4iT - 4.91e3T^{2} \)
19 \( 1 - 108.T + 6.85e3T^{2} \)
23 \( 1 + 50.2T + 1.21e4T^{2} \)
29 \( 1 - 245. iT - 2.43e4T^{2} \)
31 \( 1 + 189. iT - 2.97e4T^{2} \)
37 \( 1 - 29.0iT - 5.06e4T^{2} \)
41 \( 1 + 419. iT - 6.89e4T^{2} \)
43 \( 1 - 434. iT - 7.95e4T^{2} \)
47 \( 1 + 418.T + 1.03e5T^{2} \)
53 \( 1 - 614. iT - 1.48e5T^{2} \)
61 \( 1 - 438. iT - 2.26e5T^{2} \)
67 \( 1 + 971. iT - 3.00e5T^{2} \)
71 \( 1 - 967. iT - 3.57e5T^{2} \)
73 \( 1 + 656. iT - 3.89e5T^{2} \)
79 \( 1 - 185.T + 4.93e5T^{2} \)
83 \( 1 + 218.T + 5.71e5T^{2} \)
89 \( 1 + 385.T + 7.04e5T^{2} \)
97 \( 1 + 388. iT - 9.12e5T^{2} \)
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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

−10.98503134204253251949689280842, −9.997554320013013967370493266803, −9.138225081757981260610337994053, −8.039094975599482562853126290613, −7.21790678819585771256681219082, −6.11406014299713480650295340834, −4.90227142135928853208910128143, −3.57409112007691095330626266787, −2.90166589128836406123512158526, −1.26435022088401112200087762339, 1.60868040465392979499245888222, 2.94066856289507487389527137193, 3.89500392087362598580894388746, 4.89557702106088851397526980121, 6.47147633773043877870697578206, 7.11079447503645465356538372572, 8.300208995026574163908354158459, 9.379652367087658185055510572316, 9.947387264719542753494068769687, 11.35141334222429341052845498354

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