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Properties

Label 2-845-65.18-c1-0-34
Degree $2$
Conductor $845$
Sign $0.611 + 0.791i$
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  − 1.57i·2-s + (0.725 + 0.725i)3-s − 0.494·4-s + (−0.146 + 2.23i)5-s + (1.14 − 1.14i)6-s + 4.24·7-s − 2.37i·8-s − 1.94i·9-s + (3.52 + 0.231i)10-s + (−1.14 − 1.14i)11-s + (−0.358 − 0.358i)12-s − 6.71i·14-s + (−1.72 + 1.51i)15-s − 4.74·16-s + (4.37 + 4.37i)17-s − 3.07·18-s + ⋯
L(s)  = 1  − 1.11i·2-s + (0.419 + 0.419i)3-s − 0.247·4-s + (−0.0654 + 0.997i)5-s + (0.468 − 0.468i)6-s + 1.60·7-s − 0.840i·8-s − 0.648i·9-s + (1.11 + 0.0731i)10-s + (−0.345 − 0.345i)11-s + (−0.103 − 0.103i)12-s − 1.79i·14-s + (−0.445 + 0.390i)15-s − 1.18·16-s + (1.06 + 1.06i)17-s − 0.724·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.611 + 0.791i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.611 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.611 + 0.791i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (408, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.611 + 0.791i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.04101 - 1.00247i\)
\(L(\frac12)\) \(\approx\) \(2.04101 - 1.00247i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.146 - 2.23i)T \)
13 \( 1 \)
good2 \( 1 + 1.57iT - 2T^{2} \)
3 \( 1 + (-0.725 - 0.725i)T + 3iT^{2} \)
7 \( 1 - 4.24T + 7T^{2} \)
11 \( 1 + (1.14 + 1.14i)T + 11iT^{2} \)
17 \( 1 + (-4.37 - 4.37i)T + 17iT^{2} \)
19 \( 1 + (-0.274 - 0.274i)T + 19iT^{2} \)
23 \( 1 + (1.64 - 1.64i)T - 23iT^{2} \)
29 \( 1 - 2.79iT - 29T^{2} \)
31 \( 1 + (-2.30 + 2.30i)T - 31iT^{2} \)
37 \( 1 - 2.04T + 37T^{2} \)
41 \( 1 + (-0.883 + 0.883i)T - 41iT^{2} \)
43 \( 1 + (-0.944 + 0.944i)T - 43iT^{2} \)
47 \( 1 + 0.483T + 47T^{2} \)
53 \( 1 + (7.24 + 7.24i)T + 53iT^{2} \)
59 \( 1 + (0.311 - 0.311i)T - 59iT^{2} \)
61 \( 1 - 11.2T + 61T^{2} \)
67 \( 1 + 7.11iT - 67T^{2} \)
71 \( 1 + (7.42 - 7.42i)T - 71iT^{2} \)
73 \( 1 - 4.96iT - 73T^{2} \)
79 \( 1 - 10.7iT - 79T^{2} \)
83 \( 1 + 11.8T + 83T^{2} \)
89 \( 1 + (-1.10 + 1.10i)T - 89iT^{2} \)
97 \( 1 - 10.7iT - 97T^{2} \)
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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

−10.26095481667427306844391709290, −9.604657854813304109449621990537, −8.399007593444446615040355721234, −7.75524758072734376062354225487, −6.64833751010499902802652292361, −5.55587612265680334366261865737, −4.16351077075492279220493832716, −3.49297160018444884912137263431, −2.50716029107080190675237443384, −1.36964650613202429063113384401, 1.44859213736586901190485034746, 2.51731156255315291503009968360, 4.60247439932500332854545369920, 5.01923127781880977642701715811, 5.84866600966090916041498532076, 7.29730822906973927831143495024, 7.77510827597754500860655161209, 8.239449320360970192689508641844, 8.992930021018040640932917397256, 10.21720254401469966878329369392

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