Properties

Label 2-845-1.1-c1-0-45
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  + 2.63·2-s + 1.98·3-s + 4.94·4-s + 5-s + 5.24·6-s − 3.28·7-s + 7.77·8-s + 0.957·9-s + 2.63·10-s − 3.22·11-s + 9.84·12-s − 8.65·14-s + 1.98·15-s + 10.5·16-s − 4.25·17-s + 2.52·18-s − 2.87·19-s + 4.94·20-s − 6.52·21-s − 8.51·22-s + 6.09·23-s + 15.4·24-s + 25-s − 4.06·27-s − 16.2·28-s + 5.77·29-s + 5.24·30-s + ⋯
L(s)  = 1  + 1.86·2-s + 1.14·3-s + 2.47·4-s + 0.447·5-s + 2.14·6-s − 1.24·7-s + 2.74·8-s + 0.319·9-s + 0.833·10-s − 0.973·11-s + 2.84·12-s − 2.31·14-s + 0.513·15-s + 2.64·16-s − 1.03·17-s + 0.595·18-s − 0.659·19-s + 1.10·20-s − 1.42·21-s − 1.81·22-s + 1.27·23-s + 3.15·24-s + 0.200·25-s − 0.781·27-s − 3.06·28-s + 1.07·29-s + 0.957·30-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

\(L(1)\) \(\approx\) \(5.950882671\)
\(L(\frac12)\) \(\approx\) \(5.950882671\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.63T + 2T^{2} \)
3 \( 1 - 1.98T + 3T^{2} \)
7 \( 1 + 3.28T + 7T^{2} \)
11 \( 1 + 3.22T + 11T^{2} \)
17 \( 1 + 4.25T + 17T^{2} \)
19 \( 1 + 2.87T + 19T^{2} \)
23 \( 1 - 6.09T + 23T^{2} \)
29 \( 1 - 5.77T + 29T^{2} \)
31 \( 1 + 0.835T + 31T^{2} \)
37 \( 1 + 5.59T + 37T^{2} \)
41 \( 1 - 2.18T + 41T^{2} \)
43 \( 1 - 2.48T + 43T^{2} \)
47 \( 1 - 10.5T + 47T^{2} \)
53 \( 1 + 5.08T + 53T^{2} \)
59 \( 1 + 0.144T + 59T^{2} \)
61 \( 1 - 6.06T + 61T^{2} \)
67 \( 1 - 12.6T + 67T^{2} \)
71 \( 1 + 9.02T + 71T^{2} \)
73 \( 1 + 5.70T + 73T^{2} \)
79 \( 1 - 14.1T + 79T^{2} \)
83 \( 1 - 7.41T + 83T^{2} \)
89 \( 1 + 13.0T + 89T^{2} \)
97 \( 1 - 2.97T + 97T^{2} \)
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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

−10.42545029678338512196375500161, −9.350334929396781131633889896657, −8.451171030646510962991812977522, −7.23671863982096264869053909768, −6.57442652165715219414876639072, −5.69981998660827483901314900837, −4.71162038832045911145369612094, −3.64765533810816480567184042038, −2.81942916635934601420539355929, −2.30117736770700978261878788885, 2.30117736770700978261878788885, 2.81942916635934601420539355929, 3.64765533810816480567184042038, 4.71162038832045911145369612094, 5.69981998660827483901314900837, 6.57442652165715219414876639072, 7.23671863982096264869053909768, 8.451171030646510962991812977522, 9.350334929396781131633889896657, 10.42545029678338512196375500161

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