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Properties

Label 2-845-845.389-c1-0-9
Degree $2$
Conductor $845$
Sign $0.668 + 0.743i$
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  + (−0.941 − 0.834i)2-s + (−2.67 − 1.01i)3-s + (−0.0503 − 0.414i)4-s + (−2.16 + 0.549i)5-s + (1.67 + 3.18i)6-s + (2.04 − 2.95i)7-s + (−1.72 + 2.50i)8-s + (3.88 + 3.44i)9-s + (2.49 + 1.29i)10-s + (−2.45 − 2.76i)11-s + (−0.286 + 1.16i)12-s + (−2.59 + 2.50i)13-s + (−4.38 + 1.08i)14-s + (6.35 + 0.729i)15-s + (2.90 − 0.715i)16-s + (0.824 + 0.569i)17-s + ⋯
L(s)  = 1  + (−0.665 − 0.589i)2-s + (−1.54 − 0.585i)3-s + (−0.0251 − 0.207i)4-s + (−0.969 + 0.245i)5-s + (0.682 + 1.30i)6-s + (0.771 − 1.11i)7-s + (−0.610 + 0.884i)8-s + (1.29 + 1.14i)9-s + (0.790 + 0.408i)10-s + (−0.739 − 0.834i)11-s + (−0.0825 + 0.335i)12-s + (−0.719 + 0.694i)13-s + (−1.17 + 0.289i)14-s + (1.64 + 0.188i)15-s + (0.725 − 0.178i)16-s + (0.200 + 0.138i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.668 + 0.743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.668 + 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.668 + 0.743i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (389, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.668 + 0.743i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.278473 - 0.124023i\)
\(L(\frac12)\) \(\approx\) \(0.278473 - 0.124023i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.16 - 0.549i)T \)
13 \( 1 + (2.59 - 2.50i)T \)
good2 \( 1 + (0.941 + 0.834i)T + (0.241 + 1.98i)T^{2} \)
3 \( 1 + (2.67 + 1.01i)T + (2.24 + 1.98i)T^{2} \)
7 \( 1 + (-2.04 + 2.95i)T + (-2.48 - 6.54i)T^{2} \)
11 \( 1 + (2.45 + 2.76i)T + (-1.32 + 10.9i)T^{2} \)
17 \( 1 + (-0.824 - 0.569i)T + (6.02 + 15.8i)T^{2} \)
19 \( 1 - 6.19iT - 19T^{2} \)
23 \( 1 - 7.08iT - 23T^{2} \)
29 \( 1 + (5.65 + 5.01i)T + (3.49 + 28.7i)T^{2} \)
31 \( 1 + (-1.73 - 3.30i)T + (-17.6 + 25.5i)T^{2} \)
37 \( 1 + (-7.74 + 4.06i)T + (21.0 - 30.4i)T^{2} \)
41 \( 1 + (4.49 + 1.70i)T + (30.6 + 27.1i)T^{2} \)
43 \( 1 + (-0.0325 + 0.0620i)T + (-24.4 - 35.3i)T^{2} \)
47 \( 1 + (0.169 - 1.39i)T + (-45.6 - 11.2i)T^{2} \)
53 \( 1 + (11.1 + 7.72i)T + (18.7 + 49.5i)T^{2} \)
59 \( 1 + (-3.32 + 13.4i)T + (-52.2 - 27.4i)T^{2} \)
61 \( 1 + (0.672 + 0.974i)T + (-21.6 + 57.0i)T^{2} \)
67 \( 1 + (0.550 - 4.53i)T + (-65.0 - 16.0i)T^{2} \)
71 \( 1 + (-7.84 - 2.97i)T + (53.1 + 47.0i)T^{2} \)
73 \( 1 + (-0.428 + 0.379i)T + (8.79 - 72.4i)T^{2} \)
79 \( 1 + (-0.409 + 3.37i)T + (-76.7 - 18.9i)T^{2} \)
83 \( 1 + (-2.24 - 5.91i)T + (-62.1 + 55.0i)T^{2} \)
89 \( 1 - 16.3iT - 89T^{2} \)
97 \( 1 + (1.62 - 0.400i)T + (85.8 - 45.0i)T^{2} \)
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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

−10.41132777525165267382435690806, −9.618414355957963196769420551683, −7.954650296975851435305303756186, −7.79616066930905551353098655055, −6.67133273893507820387644413248, −5.65247151639270047215778801540, −4.90344660397165080994175063434, −3.69798630109481021354851237462, −1.79510128144454837355035944542, −0.69471874264820590543817084547, 0.39665269018145556538549540477, 2.83332659710906090233143859705, 4.57372149722853991687577706286, 4.87483748974009688197590583597, 5.91612705555293436662076643027, 7.01090536993075596591351843185, 7.71053622251721495319812050695, 8.551772653816691354946734128854, 9.401227669669680739082219981577, 10.31525986216233362053201134880

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