Properties

Label 2-845-1.1-c1-0-41
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  − 1.21·2-s + 2.33·3-s − 0.512·4-s + 5-s − 2.84·6-s − 3.60·7-s + 3.06·8-s + 2.43·9-s − 1.21·10-s − 5.37·11-s − 1.19·12-s + 4.39·14-s + 2.33·15-s − 2.71·16-s − 1.13·17-s − 2.97·18-s − 2.26·19-s − 0.512·20-s − 8.39·21-s + 6.55·22-s − 3.89·23-s + 7.14·24-s + 25-s − 1.30·27-s + 1.84·28-s − 0.0247·29-s − 2.84·30-s + ⋯
L(s)  = 1  − 0.862·2-s + 1.34·3-s − 0.256·4-s + 0.447·5-s − 1.16·6-s − 1.36·7-s + 1.08·8-s + 0.813·9-s − 0.385·10-s − 1.61·11-s − 0.344·12-s + 1.17·14-s + 0.602·15-s − 0.678·16-s − 0.274·17-s − 0.701·18-s − 0.520·19-s − 0.114·20-s − 1.83·21-s + 1.39·22-s − 0.811·23-s + 1.45·24-s + 0.200·25-s − 0.251·27-s + 0.348·28-s − 0.00459·29-s − 0.519·30-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: $\chi_{845} (1, \cdot )$
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 \( 1 + 1.21T + 2T^{2} \)
3 \( 1 - 2.33T + 3T^{2} \)
7 \( 1 + 3.60T + 7T^{2} \)
11 \( 1 + 5.37T + 11T^{2} \)
17 \( 1 + 1.13T + 17T^{2} \)
19 \( 1 + 2.26T + 19T^{2} \)
23 \( 1 + 3.89T + 23T^{2} \)
29 \( 1 + 0.0247T + 29T^{2} \)
31 \( 1 + 5.46T + 31T^{2} \)
37 \( 1 - 8.70T + 37T^{2} \)
41 \( 1 + 3.73T + 41T^{2} \)
43 \( 1 + 1.13T + 43T^{2} \)
47 \( 1 + 2.58T + 47T^{2} \)
53 \( 1 + 4.43T + 53T^{2} \)
59 \( 1 - 0.171T + 59T^{2} \)
61 \( 1 - 3.36T + 61T^{2} \)
67 \( 1 + 6.39T + 67T^{2} \)
71 \( 1 - 10.7T + 71T^{2} \)
73 \( 1 - 4.70T + 73T^{2} \)
79 \( 1 + 11.9T + 79T^{2} \)
83 \( 1 - 12.1T + 83T^{2} \)
89 \( 1 - 16.1T + 89T^{2} \)
97 \( 1 + 12.1T + 97T^{2} \)
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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

−9.625829059469435198820628255089, −9.042693443659476638159445516824, −8.167800203022292438696664529422, −7.67340167559584802764663393421, −6.55254510447461649334817236883, −5.35761659014663467352607282305, −4.04299855008197229101909617313, −2.96804483755670435285527783189, −2.08415832884535496228005583226, 0, 2.08415832884535496228005583226, 2.96804483755670435285527783189, 4.04299855008197229101909617313, 5.35761659014663467352607282305, 6.55254510447461649334817236883, 7.67340167559584802764663393421, 8.167800203022292438696664529422, 9.042693443659476638159445516824, 9.625829059469435198820628255089

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