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Properties

Label 2-845-13.12-c1-0-45
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 + 2.73·3-s − 0.999·4-s − i·5-s − 4.73i·6-s − 2i·7-s − 1.73i·8-s + 4.46·9-s − 1.73·10-s + 4.73i·11-s − 2.73·12-s − 3.46·14-s − 2.73i·15-s − 5·16-s + 3.46·17-s − 7.73i·18-s + ⋯
L(s)  = 1  − 1.22i·2-s + 1.57·3-s − 0.499·4-s − 0.447i·5-s − 1.93i·6-s − 0.755i·7-s − 0.612i·8-s + 1.48·9-s − 0.547·10-s + 1.42i·11-s − 0.788·12-s − 0.925·14-s − 0.705i·15-s − 1.25·16-s + 0.840·17-s − 1.82i·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

\(L(1)\) \(\approx\) \(1.30688 - 2.44194i\)
\(L(\frac12)\) \(\approx\) \(1.30688 - 2.44194i\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.73T + 3T^{2} \)
7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 - 4.73iT - 11T^{2} \)
17 \( 1 - 3.46T + 17T^{2} \)
19 \( 1 + 6.19iT - 19T^{2} \)
23 \( 1 + 1.26T + 23T^{2} \)
29 \( 1 + 2.53T + 29T^{2} \)
31 \( 1 - 10.1iT - 31T^{2} \)
37 \( 1 - 4iT - 37T^{2} \)
41 \( 1 - 3.46iT - 41T^{2} \)
43 \( 1 - 0.196T + 43T^{2} \)
47 \( 1 + 6iT - 47T^{2} \)
53 \( 1 - 10.3T + 53T^{2} \)
59 \( 1 + 9.12iT - 59T^{2} \)
61 \( 1 + 8.39T + 61T^{2} \)
67 \( 1 - 6.39iT - 67T^{2} \)
71 \( 1 - 4.73iT - 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 + 8.39T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 - 12.9iT - 89T^{2} \)
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

−9.941458448012231860554729679544, −9.265089086663602082161951049185, −8.499118740985588443995218732153, −7.38395192315750668002601933356, −6.93457527284559248569112368375, −4.88809001428979001219866922323, −4.05916982647421300971515370770, −3.20247770031679455055116079903, −2.26142310007952345121911542415, −1.27545013423109265229125539164, 2.09511653932071999289730720517, 3.06309547032121625227549428416, 3.99353023372760538219440792833, 5.76250675660818695144590982684, 5.97091499516259207279901280593, 7.45901204796264639705123777790, 7.83647031389198799370433746837, 8.603490921793121437602537141958, 9.161211778438458756678031615641, 10.17151001556650161346390144863

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