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Properties

Label 2-845-65.57-c1-0-40
Degree $2$
Conductor $845$
Sign $-0.291 + 0.956i$
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  − 1.83·2-s + (−1.40 + 1.40i)3-s + 1.35·4-s + (−1.45 − 1.69i)5-s + (2.56 − 2.56i)6-s − 3.53i·7-s + 1.18·8-s − 0.927i·9-s + (2.66 + 3.11i)10-s + (2.73 + 2.73i)11-s + (−1.89 + 1.89i)12-s + 6.48i·14-s + (4.41 + 0.340i)15-s − 4.87·16-s + (1.43 − 1.43i)17-s + 1.69i·18-s + ⋯
L(s)  = 1  − 1.29·2-s + (−0.809 + 0.809i)3-s + 0.677·4-s + (−0.650 − 0.759i)5-s + (1.04 − 1.04i)6-s − 1.33i·7-s + 0.417·8-s − 0.309i·9-s + (0.842 + 0.983i)10-s + (0.825 + 0.825i)11-s + (−0.548 + 0.548i)12-s + 1.73i·14-s + (1.14 + 0.0880i)15-s − 1.21·16-s + (0.347 − 0.347i)17-s + 0.400i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.291 + 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.291 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.291 + 0.956i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.291 + 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.159598 - 0.215496i\)
\(L(\frac12)\) \(\approx\) \(0.159598 - 0.215496i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.45 + 1.69i)T \)
13 \( 1 \)
good2 \( 1 + 1.83T + 2T^{2} \)
3 \( 1 + (1.40 - 1.40i)T - 3iT^{2} \)
7 \( 1 + 3.53iT - 7T^{2} \)
11 \( 1 + (-2.73 - 2.73i)T + 11iT^{2} \)
17 \( 1 + (-1.43 + 1.43i)T - 17iT^{2} \)
19 \( 1 + (0.379 + 0.379i)T + 19iT^{2} \)
23 \( 1 + (-0.215 - 0.215i)T + 23iT^{2} \)
29 \( 1 - 1.97iT - 29T^{2} \)
31 \( 1 + (4.13 - 4.13i)T - 31iT^{2} \)
37 \( 1 + 5.41iT - 37T^{2} \)
41 \( 1 + (-0.475 + 0.475i)T - 41iT^{2} \)
43 \( 1 + (-6.23 - 6.23i)T + 43iT^{2} \)
47 \( 1 + 9.75iT - 47T^{2} \)
53 \( 1 + (-3.16 + 3.16i)T - 53iT^{2} \)
59 \( 1 + (-8.59 + 8.59i)T - 59iT^{2} \)
61 \( 1 + 2.88T + 61T^{2} \)
67 \( 1 + 2.28T + 67T^{2} \)
71 \( 1 + (-3.26 + 3.26i)T - 71iT^{2} \)
73 \( 1 + 14.7T + 73T^{2} \)
79 \( 1 - 1.59iT - 79T^{2} \)
83 \( 1 + 7.57iT - 83T^{2} \)
89 \( 1 + (3.32 - 3.32i)T - 89iT^{2} \)
97 \( 1 + 17.8T + 97T^{2} \)
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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

−9.865583456348361393885565156117, −9.325449437878182071264928450579, −8.381672955347726159891023371870, −7.42804968172643607679965574117, −6.94615358203187668030557610846, −5.33073348885777554998006868110, −4.43811640185042077701227805584, −3.90433989892087524389649948136, −1.47062532236214640197545799896, −0.28193744396235291041692762393, 1.16332818795185485175966877362, 2.54687360256790516274015257941, 3.98943970837392915001305787344, 5.65819844202059852510006737833, 6.29651816321480356663529336905, 7.14815408464686477791029094299, 7.951070174867352843793347735009, 8.749359805773951719548498485963, 9.417989725149538158249236535235, 10.52430518077060987785424147497

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