Properties

Label 2-7105-1.1-c1-0-80
Degree $2$
Conductor $7105$
Sign $1$
Analytic cond. $56.7337$
Root an. cond. $7.53217$
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  + 0.311·2-s − 2.90·3-s − 1.90·4-s − 5-s − 0.903·6-s − 1.21·8-s + 5.42·9-s − 0.311·10-s − 1.52·11-s + 5.52·12-s + 0.622·13-s + 2.90·15-s + 3.42·16-s + 7.95·17-s + 1.68·18-s + 1.09·19-s + 1.90·20-s − 0.474·22-s + 7.52·23-s + 3.52·24-s + 25-s + 0.193·26-s − 7.05·27-s − 29-s + 0.903·30-s + 6.90·31-s + 3.49·32-s + ⋯
L(s)  = 1  + 0.219·2-s − 1.67·3-s − 0.951·4-s − 0.447·5-s − 0.368·6-s − 0.429·8-s + 1.80·9-s − 0.0983·10-s − 0.459·11-s + 1.59·12-s + 0.172·13-s + 0.749·15-s + 0.857·16-s + 1.92·17-s + 0.398·18-s + 0.251·19-s + 0.425·20-s − 0.101·22-s + 1.56·23-s + 0.719·24-s + 0.200·25-s + 0.0379·26-s − 1.35·27-s − 0.185·29-s + 0.164·30-s + 1.23·31-s + 0.617·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7105\)    =    \(5 \cdot 7^{2} \cdot 29\)
Sign: $1$
Analytic conductor: \(56.7337\)
Root analytic conductor: \(7.53217\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7105,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8524338106\)
\(L(\frac12)\) \(\approx\) \(0.8524338106\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + T \)
7 \( 1 \)
29 \( 1 + T \)
good2 \( 1 - 0.311T + 2T^{2} \)
3 \( 1 + 2.90T + 3T^{2} \)
11 \( 1 + 1.52T + 11T^{2} \)
13 \( 1 - 0.622T + 13T^{2} \)
17 \( 1 - 7.95T + 17T^{2} \)
19 \( 1 - 1.09T + 19T^{2} \)
23 \( 1 - 7.52T + 23T^{2} \)
31 \( 1 - 6.90T + 31T^{2} \)
37 \( 1 - 3.95T + 37T^{2} \)
41 \( 1 + 3.67T + 41T^{2} \)
43 \( 1 + 10.5T + 43T^{2} \)
47 \( 1 + 6.90T + 47T^{2} \)
53 \( 1 - 6.42T + 53T^{2} \)
59 \( 1 - 1.67T + 59T^{2} \)
61 \( 1 - 1.86T + 61T^{2} \)
67 \( 1 - 11.5T + 67T^{2} \)
71 \( 1 - 13.6T + 71T^{2} \)
73 \( 1 + 10.1T + 73T^{2} \)
79 \( 1 - 9.13T + 79T^{2} \)
83 \( 1 + 10.7T + 83T^{2} \)
89 \( 1 - 7.80T + 89T^{2} \)
97 \( 1 - 4.08T + 97T^{2} \)
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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

−7.945557574579502203729604102492, −7.07347124709586413116758430667, −6.41206504746946424421650534706, −5.54525432744848025520565310558, −5.16951855885689172223373278470, −4.68946100721651651806758754269, −3.72203777906503592039903505028, −3.04212942244126921001894949381, −1.21693149359206007935981309955, −0.60164466764709793396470255615, 0.60164466764709793396470255615, 1.21693149359206007935981309955, 3.04212942244126921001894949381, 3.72203777906503592039903505028, 4.68946100721651651806758754269, 5.16951855885689172223373278470, 5.54525432744848025520565310558, 6.41206504746946424421650534706, 7.07347124709586413116758430667, 7.945557574579502203729604102492

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