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Properties

Label 2-832-13.12-c3-0-45
Degree $2$
Conductor $832$
Sign $0.408 - 0.912i$
Analytic cond. $49.0895$
Root an. cond. $7.00639$
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  + 9.27·3-s + 2.27i·5-s + 28.5i·7-s + 59.0·9-s − 0.604i·11-s + (19.1 − 42.7i)13-s + 21.1i·15-s − 41.0·17-s + 129. i·19-s + 264. i·21-s − 73.8·23-s + 119.·25-s + 297.·27-s + 214.·29-s − 126. i·31-s + ⋯
L(s)  = 1  + 1.78·3-s + 0.203i·5-s + 1.54i·7-s + 2.18·9-s − 0.0165i·11-s + (0.408 − 0.912i)13-s + 0.364i·15-s − 0.585·17-s + 1.56i·19-s + 2.75i·21-s − 0.669·23-s + 0.958·25-s + 2.11·27-s + 1.37·29-s − 0.731i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.408 - 0.912i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.408 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(832\)    =    \(2^{6} \cdot 13\)
Sign: $0.408 - 0.912i$
Analytic conductor: \(49.0895\)
Root analytic conductor: \(7.00639\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{832} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 832,\ (\ :3/2),\ 0.408 - 0.912i)\)

Particular Values

\(L(2)\) \(\approx\) \(4.114665536\)
\(L(\frac12)\) \(\approx\) \(4.114665536\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (-19.1 + 42.7i)T \)
good3 \( 1 - 9.27T + 27T^{2} \)
5 \( 1 - 2.27iT - 125T^{2} \)
7 \( 1 - 28.5iT - 343T^{2} \)
11 \( 1 + 0.604iT - 1.33e3T^{2} \)
17 \( 1 + 41.0T + 4.91e3T^{2} \)
19 \( 1 - 129. iT - 6.85e3T^{2} \)
23 \( 1 + 73.8T + 1.21e4T^{2} \)
29 \( 1 - 214.T + 2.43e4T^{2} \)
31 \( 1 + 126. iT - 2.97e4T^{2} \)
37 \( 1 - 309. iT - 5.06e4T^{2} \)
41 \( 1 - 88.1iT - 6.89e4T^{2} \)
43 \( 1 + 245.T + 7.95e4T^{2} \)
47 \( 1 + 68.5iT - 1.03e5T^{2} \)
53 \( 1 - 613.T + 1.48e5T^{2} \)
59 \( 1 - 840. iT - 2.05e5T^{2} \)
61 \( 1 + 587.T + 2.26e5T^{2} \)
67 \( 1 + 606. iT - 3.00e5T^{2} \)
71 \( 1 - 507. iT - 3.57e5T^{2} \)
73 \( 1 + 177. iT - 3.89e5T^{2} \)
79 \( 1 + 143.T + 4.93e5T^{2} \)
83 \( 1 - 773. iT - 5.71e5T^{2} \)
89 \( 1 + 1.40e3iT - 7.04e5T^{2} \)
97 \( 1 + 468. iT - 9.12e5T^{2} \)
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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

−9.853776290220865665895246710368, −8.850548520344249534692684219771, −8.407207386837760154005792103853, −7.85151908102817917508387629719, −6.58934494473007177151674430596, −5.63600110835306939484312506194, −4.33150486544975578352257692545, −3.17375142663940143887773570585, −2.62168961934050097099764306343, −1.57291508683541684003750512111, 0.871888879718714123598411209376, 2.06797626378793877395496583179, 3.17874824717472864017214831098, 4.11781250308981528324771546285, 4.68506021509933168255486299065, 6.79912138879293298743347119541, 7.05074696127921545591395817263, 8.130148483197165966579364498517, 8.812271879007861236941796484653, 9.434850065756305541442788586433

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