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Properties

Label 2-348-29.28-c5-0-13
Degree $2$
Conductor $348$
Sign $0.293 - 0.956i$
Analytic cond. $55.8135$
Root an. cond. $7.47084$
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  + 9i·3-s + 67.4·5-s − 45.8·7-s − 81·9-s − 322. i·11-s − 100.·13-s + 607. i·15-s + 1.20e3i·17-s + 739. i·19-s − 412. i·21-s + 3.07e3·23-s + 1.42e3·25-s − 729i·27-s + (4.33e3 + 1.32e3i)29-s + 5.11e3i·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 1.20·5-s − 0.353·7-s − 0.333·9-s − 0.803i·11-s − 0.164·13-s + 0.696i·15-s + 1.01i·17-s + 0.469i·19-s − 0.204i·21-s + 1.21·23-s + 0.457·25-s − 0.192i·27-s + (0.956 + 0.293i)29-s + 0.955i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.293 - 0.956i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.293 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(348\)    =    \(2^{2} \cdot 3 \cdot 29\)
Sign: $0.293 - 0.956i$
Analytic conductor: \(55.8135\)
Root analytic conductor: \(7.47084\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{348} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 348,\ (\ :5/2),\ 0.293 - 0.956i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.391876582\)
\(L(\frac12)\) \(\approx\) \(2.391876582\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - 9iT \)
29 \( 1 + (-4.33e3 - 1.32e3i)T \)
good5 \( 1 - 67.4T + 3.12e3T^{2} \)
7 \( 1 + 45.8T + 1.68e4T^{2} \)
11 \( 1 + 322. iT - 1.61e5T^{2} \)
13 \( 1 + 100.T + 3.71e5T^{2} \)
17 \( 1 - 1.20e3iT - 1.41e6T^{2} \)
19 \( 1 - 739. iT - 2.47e6T^{2} \)
23 \( 1 - 3.07e3T + 6.43e6T^{2} \)
31 \( 1 - 5.11e3iT - 2.86e7T^{2} \)
37 \( 1 + 8.58e3iT - 6.93e7T^{2} \)
41 \( 1 - 1.54e4iT - 1.15e8T^{2} \)
43 \( 1 - 1.06e4iT - 1.47e8T^{2} \)
47 \( 1 + 2.27e4iT - 2.29e8T^{2} \)
53 \( 1 - 4.00e4T + 4.18e8T^{2} \)
59 \( 1 - 3.80e4T + 7.14e8T^{2} \)
61 \( 1 - 1.90e4iT - 8.44e8T^{2} \)
67 \( 1 - 6.05e3T + 1.35e9T^{2} \)
71 \( 1 + 6.66e4T + 1.80e9T^{2} \)
73 \( 1 - 4.10e4iT - 2.07e9T^{2} \)
79 \( 1 - 4.76e4iT - 3.07e9T^{2} \)
83 \( 1 - 5.67e4T + 3.93e9T^{2} \)
89 \( 1 - 4.54e4iT - 5.58e9T^{2} \)
97 \( 1 - 6.02e4iT - 8.58e9T^{2} \)
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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

−10.54883869195121422713889825522, −10.06372823935669044130532639418, −9.076362071338030819441946359515, −8.345934103990378104433347687761, −6.78424706333471702456144193445, −5.91712357705335936824277981017, −5.10436730323148554500566665705, −3.68872704469006664473690875396, −2.57766930990804916988432222522, −1.14992329597953690194452761403, 0.66643245163087331659053086454, 2.00294811971667095944759352475, 2.88899895446717997543778012728, 4.68983372650177214722089098052, 5.67922638028973274427511998014, 6.70292231827856528618007767741, 7.38190207233168711151741983681, 8.788357273056155402441697230430, 9.578900264480957936228966647724, 10.29390216515139830983270877182

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