Properties

Label 2-845-1.1-c1-0-18
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.49·2-s − 2.82·3-s + 4.22·4-s + 5-s + 7.05·6-s − 1.90·7-s − 5.55·8-s + 4.99·9-s − 2.49·10-s − 1.06·11-s − 11.9·12-s + 4.75·14-s − 2.82·15-s + 5.41·16-s + 0.637·17-s − 12.4·18-s − 5.73·19-s + 4.22·20-s + 5.38·21-s + 2.66·22-s + 3.81·23-s + 15.7·24-s + 25-s − 5.62·27-s − 8.05·28-s + 9.45·29-s + 7.05·30-s + ⋯
L(s)  = 1  − 1.76·2-s − 1.63·3-s + 2.11·4-s + 0.447·5-s + 2.87·6-s − 0.720·7-s − 1.96·8-s + 1.66·9-s − 0.789·10-s − 0.322·11-s − 3.44·12-s + 1.27·14-s − 0.729·15-s + 1.35·16-s + 0.154·17-s − 2.93·18-s − 1.31·19-s + 0.945·20-s + 1.17·21-s + 0.568·22-s + 0.796·23-s + 3.20·24-s + 0.200·25-s − 1.08·27-s − 1.52·28-s + 1.75·29-s + 1.28·30-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: \(1\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.49T + 2T^{2} \)
3 \( 1 + 2.82T + 3T^{2} \)
7 \( 1 + 1.90T + 7T^{2} \)
11 \( 1 + 1.06T + 11T^{2} \)
17 \( 1 - 0.637T + 17T^{2} \)
19 \( 1 + 5.73T + 19T^{2} \)
23 \( 1 - 3.81T + 23T^{2} \)
29 \( 1 - 9.45T + 29T^{2} \)
31 \( 1 - 1.46T + 31T^{2} \)
37 \( 1 + 0.757T + 37T^{2} \)
41 \( 1 + 0.267T + 41T^{2} \)
43 \( 1 - 0.637T + 43T^{2} \)
47 \( 1 - 9.44T + 47T^{2} \)
53 \( 1 + 6.99T + 53T^{2} \)
59 \( 1 + 0.741T + 59T^{2} \)
61 \( 1 - 4.19T + 61T^{2} \)
67 \( 1 + 8.09T + 67T^{2} \)
71 \( 1 + 9.76T + 71T^{2} \)
73 \( 1 - 3.71T + 73T^{2} \)
79 \( 1 + 9.31T + 79T^{2} \)
83 \( 1 + 5.11T + 83T^{2} \)
89 \( 1 - 12.5T + 89T^{2} \)
97 \( 1 - 4.22T + 97T^{2} \)
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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

−9.968835118545899550969547747504, −9.105080116997518109552901283336, −8.207598342618240121616733112498, −7.02635831294781775116190016917, −6.51600577856408276116416099403, −5.82446743820835504560652361210, −4.61727994784162460116145434129, −2.68174683659477742947105495160, −1.20619947589864816191587014973, 0, 1.20619947589864816191587014973, 2.68174683659477742947105495160, 4.61727994784162460116145434129, 5.82446743820835504560652361210, 6.51600577856408276116416099403, 7.02635831294781775116190016917, 8.207598342618240121616733112498, 9.105080116997518109552901283336, 9.968835118545899550969547747504

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