Properties

Label 2-187e2-1.1-c1-0-2
Degree $2$
Conductor $34969$
Sign $1$
Analytic cond. $279.228$
Root an. cond. $16.7101$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2·3-s − 4-s − 5-s − 2·6-s + 2·7-s − 3·8-s + 9-s − 10-s + 2·12-s + 13-s + 2·14-s + 2·15-s − 16-s + 18-s + 6·19-s + 20-s − 4·21-s − 2·23-s + 6·24-s − 4·25-s + 26-s + 4·27-s − 2·28-s − 9·29-s + 2·30-s + 2·31-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s − 1/2·4-s − 0.447·5-s − 0.816·6-s + 0.755·7-s − 1.06·8-s + 1/3·9-s − 0.316·10-s + 0.577·12-s + 0.277·13-s + 0.534·14-s + 0.516·15-s − 1/4·16-s + 0.235·18-s + 1.37·19-s + 0.223·20-s − 0.872·21-s − 0.417·23-s + 1.22·24-s − 4/5·25-s + 0.196·26-s + 0.769·27-s − 0.377·28-s − 1.67·29-s + 0.365·30-s + 0.359·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 34969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34969 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(34969\)    =    \(11^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(279.228\)
Root analytic conductor: \(16.7101\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{34969} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 34969,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.212307230\)
\(L(\frac12)\) \(\approx\) \(1.212307230\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
17 \( 1 \)
good2 \( 1 - T + p T^{2} \)
3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−14.94957312459324, −14.31954186900722, −13.93890046369923, −13.32552695366702, −12.87939382406804, −12.11733999055014, −11.86219311954761, −11.38149119306753, −11.05410159778123, −10.23904295944926, −9.675487105942043, −9.071205872937379, −8.468557745232331, −7.746917716161022, −7.406549702674985, −6.472205577522104, −5.853487031334660, −5.517475390468404, −5.016098997482942, −4.375096089483625, −3.851032061344226, −3.218956535159273, −2.266809483346228, −1.204126549700840, −0.4459317944035150, 0.4459317944035150, 1.204126549700840, 2.266809483346228, 3.218956535159273, 3.851032061344226, 4.375096089483625, 5.016098997482942, 5.517475390468404, 5.853487031334660, 6.472205577522104, 7.406549702674985, 7.746917716161022, 8.468557745232331, 9.071205872937379, 9.675487105942043, 10.23904295944926, 11.05410159778123, 11.38149119306753, 11.86219311954761, 12.11733999055014, 12.87939382406804, 13.32552695366702, 13.93890046369923, 14.31954186900722, 14.94957312459324

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