Properties

Label 2-35e2-1.1-c3-0-85
Degree $2$
Conductor $1225$
Sign $-1$
Analytic cond. $72.2773$
Root an. cond. $8.50160$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.31·2-s − 5·3-s + 10.6·4-s + 21.5·6-s − 11.3·8-s − 2·9-s + 19.7·11-s − 53.1·12-s − 71.3·13-s − 36.0·16-s + 31.3·17-s + 8.63·18-s + 136.·19-s − 85.1·22-s + 100.·23-s + 56.8·24-s + 307.·26-s + 145·27-s − 288.·29-s − 208.·31-s + 246.·32-s − 98.6·33-s − 135.·34-s − 21.2·36-s − 309.·37-s − 588.·38-s + 356.·39-s + ⋯
L(s)  = 1  − 1.52·2-s − 0.962·3-s + 1.32·4-s + 1.46·6-s − 0.502·8-s − 0.0740·9-s + 0.540·11-s − 1.27·12-s − 1.52·13-s − 0.562·16-s + 0.447·17-s + 0.113·18-s + 1.64·19-s − 0.825·22-s + 0.914·23-s + 0.483·24-s + 2.32·26-s + 1.03·27-s − 1.84·29-s − 1.21·31-s + 1.36·32-s − 0.520·33-s − 0.682·34-s − 0.0984·36-s − 1.37·37-s − 2.51·38-s + 1.46·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(72.2773\)
Root analytic conductor: \(8.50160\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1225,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good2 \( 1 + 4.31T + 8T^{2} \)
3 \( 1 + 5T + 27T^{2} \)
11 \( 1 - 19.7T + 1.33e3T^{2} \)
13 \( 1 + 71.3T + 2.19e3T^{2} \)
17 \( 1 - 31.3T + 4.91e3T^{2} \)
19 \( 1 - 136.T + 6.85e3T^{2} \)
23 \( 1 - 100.T + 1.21e4T^{2} \)
29 \( 1 + 288.T + 2.43e4T^{2} \)
31 \( 1 + 208.T + 2.97e4T^{2} \)
37 \( 1 + 309.T + 5.06e4T^{2} \)
41 \( 1 + 181.T + 6.89e4T^{2} \)
43 \( 1 - 18.2T + 7.95e4T^{2} \)
47 \( 1 - 147.T + 1.03e5T^{2} \)
53 \( 1 - 127.T + 1.48e5T^{2} \)
59 \( 1 - 322.T + 2.05e5T^{2} \)
61 \( 1 - 341.T + 2.26e5T^{2} \)
67 \( 1 - 84.3T + 3.00e5T^{2} \)
71 \( 1 + 315.T + 3.57e5T^{2} \)
73 \( 1 - 1.09e3T + 3.89e5T^{2} \)
79 \( 1 - 1.23e3T + 4.93e5T^{2} \)
83 \( 1 - 643.T + 5.71e5T^{2} \)
89 \( 1 - 1.14e3T + 7.04e5T^{2} \)
97 \( 1 + 1.41e3T + 9.12e5T^{2} \)
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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

−9.337339938333790283031126639738, −8.151896113061880260420965275303, −7.21949170855399631477235080772, −6.93849799901040264460646550767, −5.53008654493037169805523538719, −5.05343563634502871415873654645, −3.45663127516272014003248717196, −2.08833329092784406373988128802, −0.926968934824066914121227996689, 0, 0.926968934824066914121227996689, 2.08833329092784406373988128802, 3.45663127516272014003248717196, 5.05343563634502871415873654645, 5.53008654493037169805523538719, 6.93849799901040264460646550767, 7.21949170855399631477235080772, 8.151896113061880260420965275303, 9.337339938333790283031126639738

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