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Properties

Label 2-832-13.12-c3-0-52
Degree $2$
Conductor $832$
Sign $0.966 + 0.257i$
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  + 1.18·3-s + 18.9i·5-s − 9.32i·7-s − 25.5·9-s − 39.8i·11-s + (45.2 + 12.0i)13-s + 22.4i·15-s + 92.6·17-s − 128. i·19-s − 11.0i·21-s − 158.·23-s − 235.·25-s − 62.3·27-s + 126.·29-s − 189. i·31-s + ⋯
L(s)  = 1  + 0.227·3-s + 1.69i·5-s − 0.503i·7-s − 0.948·9-s − 1.09i·11-s + (0.966 + 0.257i)13-s + 0.387i·15-s + 1.32·17-s − 1.55i·19-s − 0.114i·21-s − 1.44·23-s − 1.88·25-s − 0.444·27-s + 0.810·29-s − 1.09i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.257i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.966 + 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(832\)    =    \(2^{6} \cdot 13\)
Sign: $0.966 + 0.257i$
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: \((2,\ 832,\ (\ :3/2),\ 0.966 + 0.257i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.972051814\)
\(L(\frac12)\) \(\approx\) \(1.972051814\)
\(L(\frac{5}{2})\) 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 \( 1 + (-45.2 - 12.0i)T \)
good3 \( 1 - 1.18T + 27T^{2} \)
5 \( 1 - 18.9iT - 125T^{2} \)
7 \( 1 + 9.32iT - 343T^{2} \)
11 \( 1 + 39.8iT - 1.33e3T^{2} \)
17 \( 1 - 92.6T + 4.91e3T^{2} \)
19 \( 1 + 128. iT - 6.85e3T^{2} \)
23 \( 1 + 158.T + 1.21e4T^{2} \)
29 \( 1 - 126.T + 2.43e4T^{2} \)
31 \( 1 + 189. iT - 2.97e4T^{2} \)
37 \( 1 - 53.7iT - 5.06e4T^{2} \)
41 \( 1 - 136. iT - 6.89e4T^{2} \)
43 \( 1 - 518.T + 7.95e4T^{2} \)
47 \( 1 + 309. iT - 1.03e5T^{2} \)
53 \( 1 - 149.T + 1.48e5T^{2} \)
59 \( 1 - 74.3iT - 2.05e5T^{2} \)
61 \( 1 + 99.0T + 2.26e5T^{2} \)
67 \( 1 - 435. iT - 3.00e5T^{2} \)
71 \( 1 - 827. iT - 3.57e5T^{2} \)
73 \( 1 + 981. iT - 3.89e5T^{2} \)
79 \( 1 + 299.T + 4.93e5T^{2} \)
83 \( 1 - 169. iT - 5.71e5T^{2} \)
89 \( 1 + 265. iT - 7.04e5T^{2} \)
97 \( 1 + 88.9iT - 9.12e5T^{2} \)
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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

−9.983979235656424777898949378166, −8.871211246138923106584825442784, −8.018477625842465751195081615585, −7.23536755229442068749966860358, −6.21495734144664941113811314438, −5.77210119679475170128949376587, −3.99843671990264306993353878993, −3.20663835975845757491668042776, −2.47765648229670665288375061537, −0.61775850468515684751219783595, 0.989228703799775266619171772789, 2.00858963528070589186655367502, 3.51796806633627478303673009234, 4.48839174478924319594298199447, 5.61782838368094076628168244829, 5.91584380658417709012126925906, 7.70378220559082449513480637726, 8.251233467365141618022103614204, 8.905626853820485742120992180772, 9.692317884423318412009728719517

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