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Properties

Label 2-832-52.7-c1-0-3
Degree $2$
Conductor $832$
Sign $0.533 - 0.846i$
Analytic cond. $6.64355$
Root an. cond. $2.57750$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.36 − 1.36i)3-s + (0.633 + 0.633i)5-s + (0.267 − i)7-s + (2.23 + 3.86i)9-s + (−4.73 + 1.26i)11-s + (3.23 − 1.59i)13-s + (−0.633 − 2.36i)15-s + (−5.59 + 3.23i)17-s + (3.09 + 0.830i)19-s + (−2 + 2i)21-s + (−2.36 + 4.09i)23-s − 4.19i·25-s − 4.00i·27-s + (−1.5 + 2.59i)29-s + (3.73 − 3.73i)31-s + ⋯
L(s)  = 1  + (−1.36 − 0.788i)3-s + (0.283 + 0.283i)5-s + (0.101 − 0.377i)7-s + (0.744 + 1.28i)9-s + (−1.42 + 0.382i)11-s + (0.896 − 0.443i)13-s + (−0.163 − 0.610i)15-s + (−1.35 + 0.783i)17-s + (0.710 + 0.190i)19-s + (−0.436 + 0.436i)21-s + (−0.493 + 0.854i)23-s − 0.839i·25-s − 0.769i·27-s + (−0.278 + 0.482i)29-s + (0.670 − 0.670i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.533 - 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.533 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(832\)    =    \(2^{6} \cdot 13\)
Sign: $0.533 - 0.846i$
Analytic conductor: \(6.64355\)
Root analytic conductor: \(2.57750\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{832} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 832,\ (\ :1/2),\ 0.533 - 0.846i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.548917 + 0.302941i\)
\(L(\frac12)\) \(\approx\) \(0.548917 + 0.302941i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.23 + 1.59i)T \)
good3 \( 1 + (2.36 + 1.36i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.633 - 0.633i)T + 5iT^{2} \)
7 \( 1 + (-0.267 + i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (4.73 - 1.26i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (5.59 - 3.23i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.09 - 0.830i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (2.36 - 4.09i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.73 + 3.73i)T - 31iT^{2} \)
37 \( 1 + (-2.86 - 10.6i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (3.86 - 1.03i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-1 - 1.73i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-7.73 - 7.73i)T + 47iT^{2} \)
53 \( 1 - 7.73T + 53T^{2} \)
59 \( 1 + (-0.464 + 1.73i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.59 + 4.5i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.36 - 12.5i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-2.36 - 0.633i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (6.09 - 6.09i)T - 73iT^{2} \)
79 \( 1 - 12.3iT - 79T^{2} \)
83 \( 1 + (-2.19 + 2.19i)T - 83iT^{2} \)
89 \( 1 + (-1.90 - 7.09i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-0.437 + 1.63i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.56905776234469501754943470461, −9.878214772040457316947107874886, −8.370444995412130037594570757844, −7.64921383491555513033059213511, −6.74649012842567371443288662366, −6.00650846145920164572244645303, −5.32250500447841829932095173491, −4.21922874103801157219867621943, −2.56296925648340434243643081461, −1.20535291398892866975302546881, 0.40660087288661687601838351112, 2.39191378306723175061536677328, 3.95153074090327286059396727939, 4.96866723654543807378392220498, 5.50973482323966908056885013825, 6.28792554110948242848040496863, 7.36039100459451827030183113297, 8.667672070933985001159675581802, 9.274740112214517414234903492204, 10.40377864107047625840351616772

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