Properties

Label 2-845-1.1-c1-0-37
Degree $2$
Conductor $845$
Sign $1$
Analytic cond. $6.74735$
Root an. cond. $2.59756$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.30·2-s + 3-s + 3.30·4-s + 5-s + 2.30·6-s + 7-s + 3.00·8-s − 2·9-s + 2.30·10-s − 1.60·11-s + 3.30·12-s + 2.30·14-s + 15-s + 0.302·16-s + 7.60·17-s − 4.60·18-s + 5.60·19-s + 3.30·20-s + 21-s − 3.69·22-s − 3·23-s + 3.00·24-s + 25-s − 5·27-s + 3.30·28-s − 6.21·29-s + 2.30·30-s + ⋯
L(s)  = 1  + 1.62·2-s + 0.577·3-s + 1.65·4-s + 0.447·5-s + 0.940·6-s + 0.377·7-s + 1.06·8-s − 0.666·9-s + 0.728·10-s − 0.484·11-s + 0.953·12-s + 0.615·14-s + 0.258·15-s + 0.0756·16-s + 1.84·17-s − 1.08·18-s + 1.28·19-s + 0.738·20-s + 0.218·21-s − 0.788·22-s − 0.625·23-s + 0.612·24-s + 0.200·25-s − 0.962·27-s + 0.624·28-s − 1.15·29-s + 0.420·30-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(4.741236432\)
\(L(\frac12)\) \(\approx\) \(4.741236432\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
13 \( 1 \)
good2 \( 1 - 2.30T + 2T^{2} \)
3 \( 1 - T + 3T^{2} \)
7 \( 1 - T + 7T^{2} \)
11 \( 1 + 1.60T + 11T^{2} \)
17 \( 1 - 7.60T + 17T^{2} \)
19 \( 1 - 5.60T + 19T^{2} \)
23 \( 1 + 3T + 23T^{2} \)
29 \( 1 + 6.21T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + 3.60T + 37T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 + 10.2T + 43T^{2} \)
47 \( 1 + 9.21T + 47T^{2} \)
53 \( 1 + 3.21T + 53T^{2} \)
59 \( 1 - 10.8T + 59T^{2} \)
61 \( 1 + T + 61T^{2} \)
67 \( 1 - 7T + 67T^{2} \)
71 \( 1 + 4.81T + 71T^{2} \)
73 \( 1 - 0.788T + 73T^{2} \)
79 \( 1 - 5.21T + 79T^{2} \)
83 \( 1 - 9.21T + 83T^{2} \)
89 \( 1 - 6.21T + 89T^{2} \)
97 \( 1 - 8.39T + 97T^{2} \)
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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

−10.23948127815180831698568867260, −9.491597028274481016595490014246, −8.249182346494674831196633259830, −7.58170703178011570551686526041, −6.39699950830088178096986400096, −5.41123508753474257536257669286, −5.11624718349095016440146693414, −3.56770180873369175370136759490, −3.08475022846135840260877917843, −1.86263334625763701777694191280, 1.86263334625763701777694191280, 3.08475022846135840260877917843, 3.56770180873369175370136759490, 5.11624718349095016440146693414, 5.41123508753474257536257669286, 6.39699950830088178096986400096, 7.58170703178011570551686526041, 8.249182346494674831196633259830, 9.491597028274481016595490014246, 10.23948127815180831698568867260

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