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Properties

Label 2-845-13.12-c1-0-47
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  − 2.41i·2-s − 1.41·3-s − 3.82·4-s + i·5-s + 3.41i·6-s − 4.82i·7-s + 4.41i·8-s − 0.999·9-s + 2.41·10-s − 3.41i·11-s + 5.41·12-s − 11.6·14-s − 1.41i·15-s + 2.99·16-s − 0.828·17-s + 2.41i·18-s + ⋯
L(s)  = 1  − 1.70i·2-s − 0.816·3-s − 1.91·4-s + 0.447i·5-s + 1.39i·6-s − 1.82i·7-s + 1.56i·8-s − 0.333·9-s + 0.763·10-s − 1.02i·11-s + 1.56·12-s − 3.11·14-s − 0.365i·15-s + 0.749·16-s − 0.200·17-s + 0.569i·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

\(L(1)\) \(\approx\) \(0.281708 + 0.150765i\)
\(L(\frac12)\) \(\approx\) \(0.281708 + 0.150765i\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.41iT - 2T^{2} \)
3 \( 1 + 1.41T + 3T^{2} \)
7 \( 1 + 4.82iT - 7T^{2} \)
11 \( 1 + 3.41iT - 11T^{2} \)
17 \( 1 + 0.828T + 17T^{2} \)
19 \( 1 - 0.585iT - 19T^{2} \)
23 \( 1 + 1.41T + 23T^{2} \)
29 \( 1 + 5.65T + 29T^{2} \)
31 \( 1 - 1.75iT - 31T^{2} \)
37 \( 1 - 8.48iT - 37T^{2} \)
41 \( 1 + 3.17iT - 41T^{2} \)
43 \( 1 - 11.0T + 43T^{2} \)
47 \( 1 - 4.82iT - 47T^{2} \)
53 \( 1 - 2.48T + 53T^{2} \)
59 \( 1 + 1.75iT - 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 + 2iT - 67T^{2} \)
71 \( 1 - 11.8iT - 71T^{2} \)
73 \( 1 + 8.48iT - 73T^{2} \)
79 \( 1 + 8.48T + 79T^{2} \)
83 \( 1 + 3.17iT - 83T^{2} \)
89 \( 1 + 6iT - 89T^{2} \)
97 \( 1 + 7.65iT - 97T^{2} \)
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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

−10.02498869664391126552322921650, −8.993494186611507785977355054277, −7.895747582404025460684473351989, −6.82221162198251592563136398903, −5.79294103948924174409063620893, −4.56861212672698714531854720279, −3.75724676494181573123005356296, −2.90924621206307939805658916047, −1.26387986542974038084162904967, −0.18910554330363969961057759617, 2.32128960413340616336943002632, 4.31221006717434068661731417328, 5.29475780043167250898893301307, 5.65786232687484104842294795592, 6.37310602089423954763401098121, 7.38787876365808593939091285921, 8.274675599832729048513779208471, 9.082532639963212806809024666230, 9.483293762323588189663094309274, 10.96184277883929763509152715492

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