Properties

Label 2-75e2-1.1-c1-0-36
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.63·2-s + 0.661·4-s − 1.44·7-s + 2.18·8-s − 0.797·11-s + 3.02·13-s + 2.35·14-s − 4.88·16-s − 0.282·17-s + 2.88·19-s + 1.30·22-s + 3.47·23-s − 4.93·26-s − 0.956·28-s + 7.73·29-s + 2.38·31-s + 3.60·32-s + 0.460·34-s + 4.64·37-s − 4.70·38-s − 8.42·41-s + 6.07·43-s − 0.527·44-s − 5.66·46-s + 1.16·47-s − 4.91·49-s + 2.00·52-s + ⋯
L(s)  = 1  − 1.15·2-s + 0.330·4-s − 0.546·7-s + 0.771·8-s − 0.240·11-s + 0.839·13-s + 0.630·14-s − 1.22·16-s − 0.0684·17-s + 0.661·19-s + 0.277·22-s + 0.724·23-s − 0.968·26-s − 0.180·28-s + 1.43·29-s + 0.428·31-s + 0.637·32-s + 0.0789·34-s + 0.763·37-s − 0.763·38-s − 1.31·41-s + 0.927·43-s − 0.0795·44-s − 0.835·46-s + 0.169·47-s − 0.701·49-s + 0.277·52-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5625,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9088453730\)
\(L(\frac12)\) \(\approx\) \(0.9088453730\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.63T + 2T^{2} \)
7 \( 1 + 1.44T + 7T^{2} \)
11 \( 1 + 0.797T + 11T^{2} \)
13 \( 1 - 3.02T + 13T^{2} \)
17 \( 1 + 0.282T + 17T^{2} \)
19 \( 1 - 2.88T + 19T^{2} \)
23 \( 1 - 3.47T + 23T^{2} \)
29 \( 1 - 7.73T + 29T^{2} \)
31 \( 1 - 2.38T + 31T^{2} \)
37 \( 1 - 4.64T + 37T^{2} \)
41 \( 1 + 8.42T + 41T^{2} \)
43 \( 1 - 6.07T + 43T^{2} \)
47 \( 1 - 1.16T + 47T^{2} \)
53 \( 1 + 11.7T + 53T^{2} \)
59 \( 1 + 12.2T + 59T^{2} \)
61 \( 1 + 0.868T + 61T^{2} \)
67 \( 1 - 11.3T + 67T^{2} \)
71 \( 1 + 4.63T + 71T^{2} \)
73 \( 1 - 10.8T + 73T^{2} \)
79 \( 1 + 4.73T + 79T^{2} \)
83 \( 1 - 16.6T + 83T^{2} \)
89 \( 1 - 16.2T + 89T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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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

−8.044397074537842103519702134773, −7.82070538419996796992283716746, −6.70108539458757800492992545359, −6.37445694043347563600731745458, −5.20724597119952497214853260375, −4.55029505301988929348914953665, −3.51800684620811126169063803289, −2.71246645268354462206325065143, −1.49726119232719311641769842420, −0.65292005676121091854692831745, 0.65292005676121091854692831745, 1.49726119232719311641769842420, 2.71246645268354462206325065143, 3.51800684620811126169063803289, 4.55029505301988929348914953665, 5.20724597119952497214853260375, 6.37445694043347563600731745458, 6.70108539458757800492992545359, 7.82070538419996796992283716746, 8.044397074537842103519702134773

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