Properties

Label 2-75e2-1.1-c1-0-91
Degree $2$
Conductor $5625$
Sign $-1$
Analytic cond. $44.9158$
Root an. cond. $6.70192$
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  − 2.66·2-s + 5.08·4-s − 3.46·7-s − 8.21·8-s + 3.43·11-s − 5.70·13-s + 9.23·14-s + 11.6·16-s + 1.89·17-s + 2.46·19-s − 9.13·22-s + 8.84·23-s + 15.1·26-s − 17.6·28-s − 8.98·29-s − 1.90·31-s − 14.7·32-s − 5.03·34-s − 5.09·37-s − 6.57·38-s − 6.35·41-s + 3.43·43-s + 17.4·44-s − 23.5·46-s + 8.61·47-s + 5.02·49-s − 29.0·52-s + ⋯
L(s)  = 1  − 1.88·2-s + 2.54·4-s − 1.31·7-s − 2.90·8-s + 1.03·11-s − 1.58·13-s + 2.46·14-s + 2.92·16-s + 0.458·17-s + 0.566·19-s − 1.94·22-s + 1.84·23-s + 2.97·26-s − 3.33·28-s − 1.66·29-s − 0.342·31-s − 2.60·32-s − 0.863·34-s − 0.837·37-s − 1.06·38-s − 0.991·41-s + 0.523·43-s + 2.63·44-s − 3.47·46-s + 1.25·47-s + 0.718·49-s − 4.02·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5625\)    =    \(3^{2} \cdot 5^{4}\)
Sign: $-1$
Analytic conductor: \(44.9158\)
Root analytic conductor: \(6.70192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5625} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5625,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
good2 \( 1 + 2.66T + 2T^{2} \)
7 \( 1 + 3.46T + 7T^{2} \)
11 \( 1 - 3.43T + 11T^{2} \)
13 \( 1 + 5.70T + 13T^{2} \)
17 \( 1 - 1.89T + 17T^{2} \)
19 \( 1 - 2.46T + 19T^{2} \)
23 \( 1 - 8.84T + 23T^{2} \)
29 \( 1 + 8.98T + 29T^{2} \)
31 \( 1 + 1.90T + 31T^{2} \)
37 \( 1 + 5.09T + 37T^{2} \)
41 \( 1 + 6.35T + 41T^{2} \)
43 \( 1 - 3.43T + 43T^{2} \)
47 \( 1 - 8.61T + 47T^{2} \)
53 \( 1 - 2.03T + 53T^{2} \)
59 \( 1 - 8.47T + 59T^{2} \)
61 \( 1 + 7.90T + 61T^{2} \)
67 \( 1 + 6.95T + 67T^{2} \)
71 \( 1 + 2.91T + 71T^{2} \)
73 \( 1 - 11.9T + 73T^{2} \)
79 \( 1 + 6.60T + 79T^{2} \)
83 \( 1 - 12.9T + 83T^{2} \)
89 \( 1 - 11.5T + 89T^{2} \)
97 \( 1 - 2.17T + 97T^{2} \)
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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

−7.69258938573695678529275351821, −7.13237944290590242020500130056, −6.87675616870985428978297584111, −5.97345272357160925082764802944, −5.13597473612763347075440754787, −3.63994989493359497135434513797, −2.98474425958620840376468932498, −2.09342409604575823549867468609, −1.01157731877838223240981330124, 0, 1.01157731877838223240981330124, 2.09342409604575823549867468609, 2.98474425958620840376468932498, 3.63994989493359497135434513797, 5.13597473612763347075440754787, 5.97345272357160925082764802944, 6.87675616870985428978297584111, 7.13237944290590242020500130056, 7.69258938573695678529275351821

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