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Properties

Label 2-845-65.29-c1-0-40
Degree $2$
Conductor $845$
Sign $0.224 + 0.974i$
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  + (−2.31 + 1.33i)2-s + (0.416 − 0.240i)3-s + (2.57 − 4.46i)4-s + (−1.67 + 1.48i)5-s + (−0.643 + 1.11i)6-s + (0.698 + 0.403i)7-s + 8.44i·8-s + (−1.38 + 2.39i)9-s + (1.89 − 5.67i)10-s + (−1.83 − 3.18i)11-s − 2.48i·12-s − 2.15·14-s + (−0.341 + 1.02i)15-s + (−6.13 − 10.6i)16-s + (1.16 + 0.675i)17-s − 7.40i·18-s + ⋯
L(s)  = 1  + (−1.63 + 0.945i)2-s + (0.240 − 0.138i)3-s + (1.28 − 2.23i)4-s + (−0.749 + 0.662i)5-s + (−0.262 + 0.455i)6-s + (0.263 + 0.152i)7-s + 2.98i·8-s + (−0.461 + 0.799i)9-s + (0.600 − 1.79i)10-s + (−0.554 − 0.959i)11-s − 0.716i·12-s − 0.576·14-s + (−0.0882 + 0.263i)15-s + (−1.53 − 2.65i)16-s + (0.283 + 0.163i)17-s − 1.74i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.224 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.224 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.224 + 0.974i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (484, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.224 + 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.120996 - 0.0963249i\)
\(L(\frac12)\) \(\approx\) \(0.120996 - 0.0963249i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.67 - 1.48i)T \)
13 \( 1 \)
good2 \( 1 + (2.31 - 1.33i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-0.416 + 0.240i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (-0.698 - 0.403i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.83 + 3.18i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.16 - 0.675i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.837 + 1.45i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.61 - 3.24i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.20 - 2.09i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.28T + 31T^{2} \)
37 \( 1 + (3.26 - 1.88i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.15 + 7.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.88 + 3.39i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.19iT - 47T^{2} \)
53 \( 1 + 5.73iT - 53T^{2} \)
59 \( 1 + (2.99 - 5.18i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.884 + 1.53i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.56 - 4.94i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.28 + 7.41i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 11.7iT - 73T^{2} \)
79 \( 1 - 2.26T + 79T^{2} \)
83 \( 1 - 3.84iT - 83T^{2} \)
89 \( 1 + (1.38 + 2.40i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.62 + 0.936i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.04356112045830444916673850557, −8.703096772666869793799797576353, −8.307052954550243452353843713383, −7.70137157414124846186087904136, −6.93983699244497104395391514370, −5.95979444165801217633812475375, −5.11339565872399071573703872907, −3.20484865450989944021981405624, −1.94711168077284511248016092615, −0.13437396644603396618278078017, 1.24624981829709849162695252939, 2.59653423217016225710255525410, 3.64534429294172371525966400937, 4.65941072943927357505726737497, 6.41837936192036255615372693687, 7.58410731890880423502678973821, 8.085805393352109120828664254340, 8.719415147999439793865532366129, 9.662520003148753763666553839762, 10.07854525484080901691868224027

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