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Properties

Label 2-845-13.12-c1-0-17
Degree $2$
Conductor $845$
Sign $0.960 - 0.277i$
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  − 0.219i·2-s − 1.60·3-s + 1.95·4-s − i·5-s + 0.351i·6-s + 0.332i·7-s − 0.868i·8-s − 0.439·9-s − 0.219·10-s + 5.37i·11-s − 3.12·12-s + 0.0729·14-s + 1.60i·15-s + 3.71·16-s + 5.06·17-s + 0.0965i·18-s + ⋯
L(s)  = 1  − 0.155i·2-s − 0.923·3-s + 0.975·4-s − 0.447i·5-s + 0.143i·6-s + 0.125i·7-s − 0.306i·8-s − 0.146·9-s − 0.0694·10-s + 1.61i·11-s − 0.901·12-s + 0.0195·14-s + 0.413i·15-s + 0.928·16-s + 1.22·17-s + 0.0227i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 - 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.960 - 0.277i$
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: \((2,\ 845,\ (\ :1/2),\ 0.960 - 0.277i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38903 + 0.196478i\)
\(L(\frac12)\) \(\approx\) \(1.38903 + 0.196478i\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.219iT - 2T^{2} \)
3 \( 1 + 1.60T + 3T^{2} \)
7 \( 1 - 0.332iT - 7T^{2} \)
11 \( 1 - 5.37iT - 11T^{2} \)
17 \( 1 - 5.06T + 17T^{2} \)
19 \( 1 - 2.26iT - 19T^{2} \)
23 \( 1 - 2.83T + 23T^{2} \)
29 \( 1 + 2.90T + 29T^{2} \)
31 \( 1 - 5.46iT - 31T^{2} \)
37 \( 1 + 5.97iT - 37T^{2} \)
41 \( 1 - 3.73iT - 41T^{2} \)
43 \( 1 - 5.06T + 43T^{2} \)
47 \( 1 + 8.34iT - 47T^{2} \)
53 \( 1 + 1.56T + 53T^{2} \)
59 \( 1 + 2.70iT - 59T^{2} \)
61 \( 1 - 14.1T + 61T^{2} \)
67 \( 1 - 10.3iT - 67T^{2} \)
71 \( 1 - 12.7iT - 71T^{2} \)
73 \( 1 - 9.68iT - 73T^{2} \)
79 \( 1 - 4.51T + 79T^{2} \)
83 \( 1 + 4.26iT - 83T^{2} \)
89 \( 1 + 3.22iT - 89T^{2} \)
97 \( 1 + 2.50iT - 97T^{2} \)
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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

−10.28672035244388853247549991576, −9.722693519725468939643575557144, −8.490984210488434688780803914492, −7.40702632480759367359115115733, −6.84293494883773296123130798715, −5.69949962783678897300294714051, −5.20246981619948191458135588106, −3.88062029633142284794177026169, −2.49071663220357356569433124822, −1.25239945911159375930312235016, 0.885858634118713048739602333207, 2.68411476358167449909968385957, 3.53511168831663447872343064537, 5.23131633215657358862075802315, 5.91952410418419065251481866313, 6.46697173849324919781573045817, 7.48324470648490622914014136302, 8.245552385395345890102567633446, 9.406228149481788635038863439848, 10.70402818423043139643462112461

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