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Properties

Label 2-29157-1.1-c1-0-1
Degree $2$
Conductor $29157$
Sign $-1$
Analytic cond. $232.819$
Root an. cond. $15.2584$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $3$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3-s + 2·4-s − 4·5-s − 2·6-s − 5·7-s + 9-s + 8·10-s − 6·11-s + 2·12-s − 6·13-s + 10·14-s − 4·15-s − 4·16-s − 6·17-s − 2·18-s − 7·19-s − 8·20-s − 5·21-s + 12·22-s + 11·25-s + 12·26-s + 27-s − 10·28-s − 6·29-s + 8·30-s − 4·31-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s + 4-s − 1.78·5-s − 0.816·6-s − 1.88·7-s + 1/3·9-s + 2.52·10-s − 1.80·11-s + 0.577·12-s − 1.66·13-s + 2.67·14-s − 1.03·15-s − 16-s − 1.45·17-s − 0.471·18-s − 1.60·19-s − 1.78·20-s − 1.09·21-s + 2.55·22-s + 11/5·25-s + 2.35·26-s + 0.192·27-s − 1.88·28-s − 1.11·29-s + 1.46·30-s − 0.718·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29157 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29157 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29157\)    =    \(3 \cdot 9719\)
Sign: $-1$
Analytic conductor: \(232.819\)
Root analytic conductor: \(15.2584\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((2,\ 29157,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 - T \)
9719 \( 1 - T \)
good2 \( 1 + p T + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
7 \( 1 + 5 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−16.03557026671796, −15.49022480285119, −15.17145147849797, −14.87561264573356, −13.64864403797745, −13.06225062775116, −12.83928046580532, −12.30676017138281, −11.57356316292761, −10.82522655754735, −10.46933322735018, −10.07229941230075, −9.275732872353130, −8.981405122615727, −8.321705194035769, −7.850491205693863, −7.343587549458040, −6.943577522024540, −6.460508342561864, −5.198012627167996, −4.525264297795713, −3.947926241807011, −3.106618306145700, −2.615523946255619, −1.961553824422640, 0, 0, 0, 1.961553824422640, 2.615523946255619, 3.106618306145700, 3.947926241807011, 4.525264297795713, 5.198012627167996, 6.460508342561864, 6.943577522024540, 7.343587549458040, 7.850491205693863, 8.321705194035769, 8.981405122615727, 9.275732872353130, 10.07229941230075, 10.46933322735018, 10.82522655754735, 11.57356316292761, 12.30676017138281, 12.83928046580532, 13.06225062775116, 13.64864403797745, 14.87561264573356, 15.17145147849797, 15.49022480285119, 16.03557026671796

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