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Properties

Label 2-845-65.7-c1-0-37
Degree $2$
Conductor $845$
Sign $0.839 - 0.543i$
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  + (0.866 + 0.5i)2-s + (0.366 + 1.36i)3-s + (−0.500 − 0.866i)4-s + (2 + i)5-s + (−0.366 + 1.36i)6-s + (−1 − 1.73i)7-s − 3i·8-s + (0.866 − 0.5i)9-s + (1.23 + 1.86i)10-s + (0.366 + 1.36i)11-s + (0.999 − i)12-s − 1.99i·14-s + (−0.633 + 3.09i)15-s + (0.500 − 0.866i)16-s + (1.36 + 0.366i)17-s + 18-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.211 + 0.788i)3-s + (−0.250 − 0.433i)4-s + (0.894 + 0.447i)5-s + (−0.149 + 0.557i)6-s + (−0.377 − 0.654i)7-s − 1.06i·8-s + (0.288 − 0.166i)9-s + (0.389 + 0.590i)10-s + (0.110 + 0.411i)11-s + (0.288 − 0.288i)12-s − 0.534i·14-s + (−0.163 + 0.799i)15-s + (0.125 − 0.216i)16-s + (0.331 + 0.0887i)17-s + 0.235·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.839 - 0.543i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (657, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.839 - 0.543i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.45395 + 0.725496i\)
\(L(\frac12)\) \(\approx\) \(2.45395 + 0.725496i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2 - i)T \)
13 \( 1 \)
good2 \( 1 + (-0.866 - 0.5i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-0.366 - 1.36i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (1 + 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.366 - 1.36i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.36 - 0.366i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-6.83 - 1.83i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-4.09 + 1.09i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (5 + 5i)T + 31iT^{2} \)
37 \( 1 + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (9.56 - 2.56i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-0.366 + 1.36i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 + (-5 + 5i)T - 53iT^{2} \)
59 \( 1 + (2.56 - 9.56i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-7 - 12.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.46 + 2i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.366 - 1.36i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + 10iT - 73T^{2} \)
79 \( 1 + 2iT - 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (6.83 - 1.83i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (1.73 - i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.00330345689814115180173370181, −9.751782758006219794185375006116, −8.950738579201673970620707721325, −7.29592278342521876728442913755, −6.78316521649286718765728522410, −5.69882536936759048747690989938, −5.00717456959150086359738553808, −3.96010405808115085474465899347, −3.18805179587204126384863910430, −1.33048370382117695537351313446, 1.41289348664036284252441248091, 2.58920020913594767061340126830, 3.42986486708478238690131639231, 4.97547141097405801596249248058, 5.44371472571557917579264891623, 6.63636174915279763552136641533, 7.54259700374529407597249766858, 8.525896234428663455730808003331, 9.157043015547866654615894841581, 10.00860688311320762578626365856

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