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Properties

Label 2-845-65.63-c1-0-50
Degree $2$
Conductor $845$
Sign $-0.237 + 0.971i$
Analytic cond. $6.74735$
Root an. cond. $2.59756$
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  + (−0.137 − 0.237i)2-s + (0.611 − 2.28i)3-s + (0.962 − 1.66i)4-s + (1.69 + 1.45i)5-s + (−0.627 + 0.168i)6-s + (0.334 + 0.193i)7-s − 1.07·8-s + (−2.23 − 1.29i)9-s + (0.112 − 0.604i)10-s + (4.21 + 1.12i)11-s + (−3.21 − 3.21i)12-s − 0.106i·14-s + (4.35 − 2.98i)15-s + (−1.77 − 3.07i)16-s + (−1.90 + 0.510i)17-s + 0.710i·18-s + ⋯
L(s)  = 1  + (−0.0971 − 0.168i)2-s + (0.353 − 1.31i)3-s + (0.481 − 0.833i)4-s + (0.759 + 0.650i)5-s + (−0.256 + 0.0686i)6-s + (0.126 + 0.0729i)7-s − 0.381·8-s + (−0.745 − 0.430i)9-s + (0.0356 − 0.191i)10-s + (1.27 + 0.340i)11-s + (−0.928 − 0.928i)12-s − 0.0283i·14-s + (1.12 − 0.771i)15-s + (−0.444 − 0.769i)16-s + (−0.462 + 0.123i)17-s + 0.167i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.237 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.237 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.237 + 0.971i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (258, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.237 + 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36235 - 1.73603i\)
\(L(\frac12)\) \(\approx\) \(1.36235 - 1.73603i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.69 - 1.45i)T \)
13 \( 1 \)
good2 \( 1 + (0.137 + 0.237i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.611 + 2.28i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (-0.334 - 0.193i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-4.21 - 1.12i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.90 - 0.510i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.29 + 4.83i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.322 + 0.0863i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-7.07 + 4.08i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.54 + 2.54i)T + 31iT^{2} \)
37 \( 1 + (4.17 - 2.41i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.20 - 4.49i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-1.76 - 6.58i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 - 9.83iT - 47T^{2} \)
53 \( 1 + (7.17 + 7.17i)T + 53iT^{2} \)
59 \( 1 + (2.34 - 0.628i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (5.32 - 9.22i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.18 - 5.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.20 + 1.12i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 - 6.08T + 73T^{2} \)
79 \( 1 - 3.34iT - 79T^{2} \)
83 \( 1 - 5.18iT - 83T^{2} \)
89 \( 1 + (-1.29 + 4.82i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (7.37 - 12.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.844261357530586674884388148844, −9.259665300066284837485172893167, −8.199104822976977027367351949936, −6.99581695931748720793388325363, −6.62395952954203989486898856464, −6.03703504765412887377169228081, −4.66156402332962929096616526744, −2.87318597791503193531854909952, −2.05441249413196440234872548205, −1.19853331229559434048470032974, 1.80793223431725146496982469510, 3.27743587114677822804621595826, 4.01262995199187825408598086235, 4.93796016354784491589561106930, 6.09413605633198623662852498594, 6.93193921853631595791713241633, 8.285088348182308208670433131168, 8.849054829090313782142440570722, 9.390187052216165872473830931763, 10.37138152539803138024712694751

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