Properties

Label 2-95e2-1.1-c1-0-209
Degree $2$
Conductor $9025$
Sign $-1$
Analytic cond. $72.0649$
Root an. cond. $8.48910$
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  − 1.82·2-s − 2.31·3-s + 1.32·4-s + 4.21·6-s + 1.45·7-s + 1.23·8-s + 2.35·9-s − 3.89·11-s − 3.05·12-s − 3.05·13-s − 2.64·14-s − 4.89·16-s + 3.92·17-s − 4.29·18-s − 3.35·21-s + 7.10·22-s − 5.37·23-s − 2.86·24-s + 5.57·26-s + 1.48·27-s + 1.91·28-s − 6·29-s + 8.43·31-s + 6.45·32-s + 9.01·33-s − 7.14·34-s + 3.11·36-s + ⋯
L(s)  = 1  − 1.28·2-s − 1.33·3-s + 0.660·4-s + 1.72·6-s + 0.548·7-s + 0.437·8-s + 0.785·9-s − 1.17·11-s − 0.883·12-s − 0.848·13-s − 0.706·14-s − 1.22·16-s + 0.951·17-s − 1.01·18-s − 0.732·21-s + 1.51·22-s − 1.12·23-s − 0.584·24-s + 1.09·26-s + 0.286·27-s + 0.362·28-s − 1.11·29-s + 1.51·31-s + 1.14·32-s + 1.56·33-s − 1.22·34-s + 0.519·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9025\)    =    \(5^{2} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(72.0649\)
Root analytic conductor: \(8.48910\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{9025} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 9025,\ (\ :1/2),\ -1)\)

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 \( 1 \)
19 \( 1 \)
good2 \( 1 + 1.82T + 2T^{2} \)
3 \( 1 + 2.31T + 3T^{2} \)
7 \( 1 - 1.45T + 7T^{2} \)
11 \( 1 + 3.89T + 11T^{2} \)
13 \( 1 + 3.05T + 13T^{2} \)
17 \( 1 - 3.92T + 17T^{2} \)
23 \( 1 + 5.37T + 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 8.43T + 31T^{2} \)
37 \( 1 - 5.95T + 37T^{2} \)
41 \( 1 + 10.4T + 41T^{2} \)
43 \( 1 + 1.45T + 43T^{2} \)
47 \( 1 - 4.90T + 47T^{2} \)
53 \( 1 - 4.23T + 53T^{2} \)
59 \( 1 + 3.35T + 59T^{2} \)
61 \( 1 - 10.3T + 61T^{2} \)
67 \( 1 - 9.84T + 67T^{2} \)
71 \( 1 + 8.64T + 71T^{2} \)
73 \( 1 + 2.43T + 73T^{2} \)
79 \( 1 - 12.4T + 79T^{2} \)
83 \( 1 - 12.6T + 83T^{2} \)
89 \( 1 + 12.3T + 89T^{2} \)
97 \( 1 + 3.05T + 97T^{2} \)
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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

−7.60457371682369263554301906337, −6.85776932720013198151242173432, −6.07882376097870575960751767145, −5.24369690173292335296227587605, −4.96828158357513067066887561296, −4.04621034445874081826162491062, −2.70261483847247809347164966736, −1.83024407632055343428133975340, −0.806976729640905341138530580520, 0, 0.806976729640905341138530580520, 1.83024407632055343428133975340, 2.70261483847247809347164966736, 4.04621034445874081826162491062, 4.96828158357513067066887561296, 5.24369690173292335296227587605, 6.07882376097870575960751767145, 6.85776932720013198151242173432, 7.60457371682369263554301906337

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