Properties

Label 2-29575-1.1-c1-0-12
Degree $2$
Conductor $29575$
Sign $-1$
Analytic cond. $236.157$
Root an. cond. $15.3674$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s − 4-s − 6-s − 7-s − 3·8-s − 2·9-s − 4·11-s + 12-s − 14-s − 16-s + 7·17-s − 2·18-s − 6·19-s + 21-s − 4·22-s + 3·24-s + 5·27-s + 28-s − 29-s + 2·31-s + 5·32-s + 4·33-s + 7·34-s + 2·36-s − 4·37-s − 6·38-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 0.377·7-s − 1.06·8-s − 2/3·9-s − 1.20·11-s + 0.288·12-s − 0.267·14-s − 1/4·16-s + 1.69·17-s − 0.471·18-s − 1.37·19-s + 0.218·21-s − 0.852·22-s + 0.612·24-s + 0.962·27-s + 0.188·28-s − 0.185·29-s + 0.359·31-s + 0.883·32-s + 0.696·33-s + 1.20·34-s + 1/3·36-s − 0.657·37-s − 0.973·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29575\)    =    \(5^{2} \cdot 7 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(236.157\)
Root analytic conductor: \(15.3674\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 29575,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 + T \)
13 \( 1 \)
good2 \( 1 - T + p T^{2} \)
3 \( 1 + T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 5 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 - 9 T + p T^{2} \)
83 \( 1 + 5 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
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

−15.29486480841818, −14.83621737637754, −14.33137304259446, −13.78769851522391, −13.22378393889962, −12.85596894759802, −12.20649591613081, −11.94950833106387, −11.26774576950370, −10.50615507576961, −10.12887939850003, −9.682978123217032, −8.641907800849525, −8.458116795083976, −7.857623636946168, −6.892612837166375, −6.406649564327144, −5.699511703732277, −5.237185179990350, −5.010605325703916, −4.012067766132286, −3.412213014431586, −2.882465295258751, −2.053845298332588, −0.7253466823521542, 0, 0.7253466823521542, 2.053845298332588, 2.882465295258751, 3.412213014431586, 4.012067766132286, 5.010605325703916, 5.237185179990350, 5.699511703732277, 6.406649564327144, 6.892612837166375, 7.857623636946168, 8.458116795083976, 8.641907800849525, 9.682978123217032, 10.12887939850003, 10.50615507576961, 11.26774576950370, 11.94950833106387, 12.20649591613081, 12.85596894759802, 13.22378393889962, 13.78769851522391, 14.33137304259446, 14.83621737637754, 15.29486480841818

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