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Properties

Label 2-845-13.12-c1-0-0
Degree $2$
Conductor $845$
Sign $-0.554 + 0.832i$
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  + 1.73i·2-s − 0.732·3-s − 0.999·4-s − i·5-s − 1.26i·6-s − 2i·7-s + 1.73i·8-s − 2.46·9-s + 1.73·10-s + 1.26i·11-s + 0.732·12-s + 3.46·14-s + 0.732i·15-s − 5·16-s − 3.46·17-s − 4.26i·18-s + ⋯
L(s)  = 1  + 1.22i·2-s − 0.422·3-s − 0.499·4-s − 0.447i·5-s − 0.517i·6-s − 0.755i·7-s + 0.612i·8-s − 0.821·9-s + 0.547·10-s + 0.382i·11-s + 0.211·12-s + 0.925·14-s + 0.189i·15-s − 1.25·16-s − 0.840·17-s − 1.00i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.554 + 0.832i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (506, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.554 + 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.107091 - 0.200103i\)
\(L(\frac12)\) \(\approx\) \(0.107091 - 0.200103i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + iT \)
13 \( 1 \)
good2 \( 1 - 1.73iT - 2T^{2} \)
3 \( 1 + 0.732T + 3T^{2} \)
7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 - 1.26iT - 11T^{2} \)
17 \( 1 + 3.46T + 17T^{2} \)
19 \( 1 - 4.19iT - 19T^{2} \)
23 \( 1 + 4.73T + 23T^{2} \)
29 \( 1 + 9.46T + 29T^{2} \)
31 \( 1 + 0.196iT - 31T^{2} \)
37 \( 1 - 4iT - 37T^{2} \)
41 \( 1 + 3.46iT - 41T^{2} \)
43 \( 1 + 10.1T + 43T^{2} \)
47 \( 1 + 6iT - 47T^{2} \)
53 \( 1 + 10.3T + 53T^{2} \)
59 \( 1 - 15.1iT - 59T^{2} \)
61 \( 1 - 12.3T + 61T^{2} \)
67 \( 1 + 14.3iT - 67T^{2} \)
71 \( 1 - 1.26iT - 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 - 12.3T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 + 0.928iT - 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−10.77076617901644991919495409457, −9.784881937825748924204099781934, −8.741062981220666006222319284835, −8.065161102080887589157173470829, −7.26747374160041593990663610613, −6.40424785885652921632956229655, −5.66656822945540211532878949214, −4.85998377840211082797309076469, −3.77417954179283042216468986387, −2.00017951303479350215762167599, 0.10573715502671083829674149249, 2.00758175216294599107049326855, 2.82776266815276402986537956134, 3.81673343901394822518910679946, 5.11433775261458911609107657041, 6.10268196564694529031558416917, 6.85873468206045005010537188332, 8.172637629122216872782670222154, 9.108029127583332264930495529156, 9.759133439372466588441662171045

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