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Properties

Label 2-845-13.12-c1-0-49
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  − 1.49i·2-s + 0.0947·3-s − 0.236·4-s − i·5-s − 0.141i·6-s − 4.82i·7-s − 2.63i·8-s − 2.99·9-s − 1.49·10-s + 1.06i·11-s − 0.0224·12-s − 7.21·14-s − 0.0947i·15-s − 4.41·16-s − 3.55·17-s + 4.47i·18-s + ⋯
L(s)  = 1  − 1.05i·2-s + 0.0547·3-s − 0.118·4-s − 0.447i·5-s − 0.0578i·6-s − 1.82i·7-s − 0.932i·8-s − 0.997·9-s − 0.472·10-s + 0.322i·11-s − 0.00647·12-s − 1.92·14-s − 0.0244i·15-s − 1.10·16-s − 0.863·17-s + 1.05i·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

\(L(1)\) \(\approx\) \(0.181026 + 1.27979i\)
\(L(\frac12)\) \(\approx\) \(0.181026 + 1.27979i\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.49iT - 2T^{2} \)
3 \( 1 - 0.0947T + 3T^{2} \)
7 \( 1 + 4.82iT - 7T^{2} \)
11 \( 1 - 1.06iT - 11T^{2} \)
17 \( 1 + 3.55T + 17T^{2} \)
19 \( 1 - 5.73iT - 19T^{2} \)
23 \( 1 - 7.08T + 23T^{2} \)
29 \( 1 - 1.47T + 29T^{2} \)
31 \( 1 + 1.46iT - 31T^{2} \)
37 \( 1 - 0.0253iT - 37T^{2} \)
41 \( 1 - 0.267iT - 41T^{2} \)
43 \( 1 + 3.55T + 43T^{2} \)
47 \( 1 + 6.51iT - 47T^{2} \)
53 \( 1 - 0.991T + 53T^{2} \)
59 \( 1 + 8.72iT - 59T^{2} \)
61 \( 1 - 6.33T + 61T^{2} \)
67 \( 1 - 5.17iT - 67T^{2} \)
71 \( 1 + 7.76iT - 71T^{2} \)
73 \( 1 + 10.1iT - 73T^{2} \)
79 \( 1 - 8.78T + 79T^{2} \)
83 \( 1 + 0.725iT - 83T^{2} \)
89 \( 1 + 13.5iT - 89T^{2} \)
97 \( 1 + 3.43iT - 97T^{2} \)
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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

−10.02639576197352425054259972035, −9.130736330685727314312927801000, −8.099424981072510453566005050533, −7.16538425680308200119733362935, −6.38841461213621657812969731219, −4.94593360939070035058171755519, −3.98444516613066405255947656654, −3.21533813449926972321094078014, −1.81961246610896341326172451469, −0.59575703449573000515754443669, 2.42885529897428974126986166635, 2.90692093427453831961686543445, 4.93172705046172383176664417781, 5.55518286951247795570842870801, 6.36845193192935445948287711674, 7.00448112248462992424165973741, 8.243706565640166221665493760689, 8.774815920281688619817472988804, 9.309747069148141155372327338461, 11.00365875367420705394357834066

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