Properties

Label 2-845-1.1-c1-0-8
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  + 0.571·2-s − 0.428·3-s − 1.67·4-s − 5-s − 0.244·6-s − 2.67·7-s − 2.10·8-s − 2.81·9-s − 0.571·10-s + 5.10·11-s + 0.715·12-s − 1.52·14-s + 0.428·15-s + 2.14·16-s + 5.34·17-s − 1.61·18-s + 6.24·19-s + 1.67·20-s + 1.14·21-s + 2.91·22-s − 2.42·23-s + 0.899·24-s + 25-s + 2.48·27-s + 4.47·28-s + 2.67·29-s + 0.244·30-s + ⋯
L(s)  = 1  + 0.404·2-s − 0.247·3-s − 0.836·4-s − 0.447·5-s − 0.0999·6-s − 1.01·7-s − 0.742·8-s − 0.938·9-s − 0.180·10-s + 1.53·11-s + 0.206·12-s − 0.408·14-s + 0.110·15-s + 0.535·16-s + 1.29·17-s − 0.379·18-s + 1.43·19-s + 0.374·20-s + 0.249·21-s + 0.622·22-s − 0.506·23-s + 0.183·24-s + 0.200·25-s + 0.479·27-s + 0.844·28-s + 0.496·29-s + 0.0446·30-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

\(L(1)\) \(\approx\) \(1.084294797\)
\(L(\frac12)\) \(\approx\) \(1.084294797\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.571T + 2T^{2} \)
3 \( 1 + 0.428T + 3T^{2} \)
7 \( 1 + 2.67T + 7T^{2} \)
11 \( 1 - 5.10T + 11T^{2} \)
17 \( 1 - 5.34T + 17T^{2} \)
19 \( 1 - 6.24T + 19T^{2} \)
23 \( 1 + 2.42T + 23T^{2} \)
29 \( 1 - 2.67T + 29T^{2} \)
31 \( 1 + 0.244T + 31T^{2} \)
37 \( 1 - 3.32T + 37T^{2} \)
41 \( 1 - 6.48T + 41T^{2} \)
43 \( 1 + 10.9T + 43T^{2} \)
47 \( 1 - 2.67T + 47T^{2} \)
53 \( 1 + 4.20T + 53T^{2} \)
59 \( 1 + 0.899T + 59T^{2} \)
61 \( 1 + 5.81T + 61T^{2} \)
67 \( 1 - 2.18T + 67T^{2} \)
71 \( 1 - 6.24T + 71T^{2} \)
73 \( 1 - 10.9T + 73T^{2} \)
79 \( 1 + 3.63T + 79T^{2} \)
83 \( 1 - 9.81T + 83T^{2} \)
89 \( 1 - 7.63T + 89T^{2} \)
97 \( 1 + 11.3T + 97T^{2} \)
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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

−9.878040743508156970089648904625, −9.477912814594913797584144726510, −8.613885669713197378538206064110, −7.68188664838355884668687872510, −6.45443824279550281763599819259, −5.82539374951588811456326573411, −4.84284398112100779238881174821, −3.62294164650344838311547739068, −3.20237830788271829994872261888, −0.816340133708459579946117519817, 0.816340133708459579946117519817, 3.20237830788271829994872261888, 3.62294164650344838311547739068, 4.84284398112100779238881174821, 5.82539374951588811456326573411, 6.45443824279550281763599819259, 7.68188664838355884668687872510, 8.613885669713197378538206064110, 9.477912814594913797584144726510, 9.878040743508156970089648904625

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