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Properties

Label 2-845-65.58-c1-0-20
Degree $2$
Conductor $845$
Sign $0.524 + 0.851i$
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  + (−0.237 − 0.137i)2-s + (−2.28 + 0.611i)3-s + (−0.962 − 1.66i)4-s + (−1.45 + 1.69i)5-s + (0.627 + 0.168i)6-s + (0.193 + 0.334i)7-s + 1.07i·8-s + (2.23 − 1.29i)9-s + (0.579 − 0.204i)10-s + (−4.21 + 1.12i)11-s + (3.21 + 3.21i)12-s − 0.106i·14-s + (2.27 − 4.76i)15-s + (−1.77 + 3.07i)16-s + (−0.510 + 1.90i)17-s − 0.710·18-s + ⋯
L(s)  = 1  + (−0.168 − 0.0971i)2-s + (−1.31 + 0.353i)3-s + (−0.481 − 0.833i)4-s + (−0.650 + 0.759i)5-s + (0.256 + 0.0686i)6-s + (0.0729 + 0.126i)7-s + 0.381i·8-s + (0.745 − 0.430i)9-s + (0.183 − 0.0646i)10-s + (−1.27 + 0.340i)11-s + (0.928 + 0.928i)12-s − 0.0283i·14-s + (0.588 − 1.23i)15-s + (−0.444 + 0.769i)16-s + (−0.123 + 0.462i)17-s − 0.167·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.524 + 0.851i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.524 + 0.851i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.254452 - 0.142169i\)
\(L(\frac12)\) \(\approx\) \(0.254452 - 0.142169i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.237 + 0.137i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (2.28 - 0.611i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (-0.193 - 0.334i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.21 - 1.12i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (0.510 - 1.90i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.29 - 4.83i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.0863 + 0.322i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (7.07 + 4.08i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.54 + 2.54i)T - 31iT^{2} \)
37 \( 1 + (-2.41 + 4.17i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.20 - 4.49i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-6.58 - 1.76i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 - 9.83T + 47T^{2} \)
53 \( 1 + (7.17 + 7.17i)T + 53iT^{2} \)
59 \( 1 + (2.34 + 0.628i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (5.32 + 9.22i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.52 - 3.18i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.20 + 1.12i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + 6.08iT - 73T^{2} \)
79 \( 1 - 3.34iT - 79T^{2} \)
83 \( 1 + 5.18T + 83T^{2} \)
89 \( 1 + (-1.29 - 4.82i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-12.7 + 7.37i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.27320014481770606892643661250, −9.627167097014538101821854948820, −8.275803859376179084620414570607, −7.51231019763541148413928751738, −6.19957294469356536161134163646, −5.76657070983033901009771032426, −4.79423421993509313363342704532, −3.94911025052450711275673623281, −2.23418103201486542798163854047, −0.28939902591876292627098520729, 0.73880104053459999836669802110, 2.91275554058756118475190861909, 4.30238653434994078869048591137, 5.00231808125529356657678931733, 5.80943426819131381614536039882, 7.18955387632601074076856390338, 7.57034740651469986098820559455, 8.637886520383381355598555045723, 9.254941534997127239608870681345, 10.65244152577487062182714819317

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