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Properties

Label 2-845-65.64-c1-0-19
Degree $2$
Conductor $845$
Sign $0.0752 - 0.997i$
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  + 0.456·2-s + i·3-s − 1.79·4-s + (2.18 − 0.456i)5-s + 0.456i·6-s − 1.73·7-s − 1.73·8-s + 2·9-s + (0.999 − 0.208i)10-s + 2.64i·11-s − 1.79i·12-s − 0.791·14-s + (0.456 + 2.18i)15-s + 2.79·16-s + 4.58i·17-s + 0.913·18-s + ⋯
L(s)  = 1  + 0.323·2-s + 0.577i·3-s − 0.895·4-s + (0.978 − 0.204i)5-s + 0.186i·6-s − 0.654·7-s − 0.612·8-s + 0.666·9-s + (0.316 − 0.0660i)10-s + 0.797i·11-s − 0.517i·12-s − 0.211·14-s + (0.117 + 0.565i)15-s + 0.697·16-s + 1.11i·17-s + 0.215·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0752 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0752 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.0752 - 0.997i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (844, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.0752 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11762 + 1.03650i\)
\(L(\frac12)\) \(\approx\) \(1.11762 + 1.03650i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.18 + 0.456i)T \)
13 \( 1 \)
good2 \( 1 - 0.456T + 2T^{2} \)
3 \( 1 - iT - 3T^{2} \)
7 \( 1 + 1.73T + 7T^{2} \)
11 \( 1 - 2.64iT - 11T^{2} \)
17 \( 1 - 4.58iT - 17T^{2} \)
19 \( 1 + 1.73iT - 19T^{2} \)
23 \( 1 - 4.58iT - 23T^{2} \)
29 \( 1 + 4.58T + 29T^{2} \)
31 \( 1 - 6.20iT - 31T^{2} \)
37 \( 1 - 7.93T + 37T^{2} \)
41 \( 1 + 2.64iT - 41T^{2} \)
43 \( 1 - 10.5iT - 43T^{2} \)
47 \( 1 + 1.82T + 47T^{2} \)
53 \( 1 - 7.58iT - 53T^{2} \)
59 \( 1 + 13.9iT - 59T^{2} \)
61 \( 1 + 1.41T + 61T^{2} \)
67 \( 1 + 1.00T + 67T^{2} \)
71 \( 1 - 7.02iT - 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 - 6T + 79T^{2} \)
83 \( 1 - 6.01T + 83T^{2} \)
89 \( 1 + 9.57iT - 89T^{2} \)
97 \( 1 + 11.4T + 97T^{2} \)
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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

−10.06804532257453675062706564701, −9.568231082351102700513115752906, −9.139337372697513665870987119530, −7.956824920381114473427643723266, −6.74164295808197742418154359568, −5.84670764953577339628399460633, −4.95383975774392076887034090459, −4.21156883632007089400852189983, −3.19219821469185835583537963155, −1.56423354552811288146131388238, 0.73148004323355908843141410694, 2.37181240413483476804350554046, 3.51562594993968745302795596177, 4.66114247935008497009805753899, 5.73182267463606994188102498417, 6.33632317702226623157225073193, 7.31035736228944956156636663733, 8.364167529736712737148235781411, 9.370930518990149022038324996215, 9.757523454534940811640514448822

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