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Properties

Label 2-845-845.389-c1-0-86
Degree $2$
Conductor $845$
Sign $-0.943 - 0.331i$
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  + (0.0655 + 0.0580i)2-s + (0.0139 + 0.00528i)3-s + (−0.240 − 1.97i)4-s + (1.39 − 1.75i)5-s + (0.000606 + 0.00115i)6-s + (−2.84 + 4.11i)7-s + (0.198 − 0.287i)8-s + (−2.24 − 1.98i)9-s + (0.192 − 0.0339i)10-s + (−0.433 − 0.489i)11-s + (0.00710 − 0.0288i)12-s + (−2.77 + 2.30i)13-s + (−0.425 + 0.104i)14-s + (0.0286 − 0.0170i)15-s + (−3.83 + 0.946i)16-s + (−2.82 − 1.94i)17-s + ⋯
L(s)  = 1  + (0.0463 + 0.0410i)2-s + (0.00804 + 0.00304i)3-s + (−0.120 − 0.988i)4-s + (0.622 − 0.782i)5-s + (0.000247 + 0.000471i)6-s + (−1.07 + 1.55i)7-s + (0.0702 − 0.101i)8-s + (−0.748 − 0.663i)9-s + (0.0610 − 0.0107i)10-s + (−0.130 − 0.147i)11-s + (0.00205 − 0.00831i)12-s + (−0.769 + 0.638i)13-s + (−0.113 + 0.0280i)14-s + (0.00739 − 0.00439i)15-s + (−0.959 + 0.236i)16-s + (−0.684 − 0.472i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 - 0.331i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.943 - 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.943 - 0.331i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (389, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.943 - 0.331i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0271267 + 0.159170i\)
\(L(\frac12)\) \(\approx\) \(0.0271267 + 0.159170i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.39 + 1.75i)T \)
13 \( 1 + (2.77 - 2.30i)T \)
good2 \( 1 + (-0.0655 - 0.0580i)T + (0.241 + 1.98i)T^{2} \)
3 \( 1 + (-0.0139 - 0.00528i)T + (2.24 + 1.98i)T^{2} \)
7 \( 1 + (2.84 - 4.11i)T + (-2.48 - 6.54i)T^{2} \)
11 \( 1 + (0.433 + 0.489i)T + (-1.32 + 10.9i)T^{2} \)
17 \( 1 + (2.82 + 1.94i)T + (6.02 + 15.8i)T^{2} \)
19 \( 1 - 5.04iT - 19T^{2} \)
23 \( 1 + 4.31iT - 23T^{2} \)
29 \( 1 + (3.06 + 2.71i)T + (3.49 + 28.7i)T^{2} \)
31 \( 1 + (-2.55 - 4.87i)T + (-17.6 + 25.5i)T^{2} \)
37 \( 1 + (5.59 - 2.93i)T + (21.0 - 30.4i)T^{2} \)
41 \( 1 + (-6.62 - 2.51i)T + (30.6 + 27.1i)T^{2} \)
43 \( 1 + (3.51 - 6.70i)T + (-24.4 - 35.3i)T^{2} \)
47 \( 1 + (-0.796 + 6.56i)T + (-45.6 - 11.2i)T^{2} \)
53 \( 1 + (5.03 + 3.47i)T + (18.7 + 49.5i)T^{2} \)
59 \( 1 + (-0.418 + 1.69i)T + (-52.2 - 27.4i)T^{2} \)
61 \( 1 + (-4.09 - 5.92i)T + (-21.6 + 57.0i)T^{2} \)
67 \( 1 + (1.04 - 8.59i)T + (-65.0 - 16.0i)T^{2} \)
71 \( 1 + (12.7 + 4.83i)T + (53.1 + 47.0i)T^{2} \)
73 \( 1 + (-1.78 + 1.58i)T + (8.79 - 72.4i)T^{2} \)
79 \( 1 + (-1.91 + 15.7i)T + (-76.7 - 18.9i)T^{2} \)
83 \( 1 + (3.26 + 8.61i)T + (-62.1 + 55.0i)T^{2} \)
89 \( 1 + 7.49iT - 89T^{2} \)
97 \( 1 + (1.97 - 0.485i)T + (85.8 - 45.0i)T^{2} \)
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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

−9.638765314610440645389768970647, −8.986797048215716732715232306455, −8.526496863363328078564969921476, −6.68321784033998883572808390622, −6.07775641151365121622316118522, −5.50510692299759876047328032367, −4.57814409064526361658273113516, −2.90919978290008411615247973797, −1.92119721079402843590215941608, −0.06983023175154067064220101340, 2.43196197997267360805155841728, 3.20855966971910690826507050942, 4.14895222437623076133433197602, 5.38363084637583412651084689884, 6.62260906670822425200146711603, 7.25663413023233824277296823820, 7.84339396816932962415408813675, 9.128158730055848195413985929520, 9.829313200252141765349168656604, 10.82955822534405750085392233911

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