Properties

Label 2-15e2-1.1-c7-0-6
Degree $2$
Conductor $225$
Sign $1$
Analytic cond. $70.2866$
Root an. cond. $8.38371$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·2-s − 103·4-s − 930·7-s + 1.15e3·8-s + 8.45e3·11-s − 6.22e3·13-s + 4.65e3·14-s + 7.40e3·16-s − 9.59e3·17-s − 4.58e4·19-s − 4.22e4·22-s − 1.02e5·23-s + 3.11e4·26-s + 9.57e4·28-s + 8.75e4·29-s − 7.62e4·31-s − 1.84e5·32-s + 4.79e4·34-s − 2.64e5·37-s + 2.29e5·38-s + 1.03e5·41-s + 3.24e5·43-s − 8.70e5·44-s + 5.10e5·46-s + 8.55e5·47-s + 4.13e4·49-s + 6.40e5·52-s + ⋯
L(s)  = 1  − 0.441·2-s − 0.804·4-s − 1.02·7-s + 0.797·8-s + 1.91·11-s − 0.785·13-s + 0.452·14-s + 0.452·16-s − 0.473·17-s − 1.53·19-s − 0.845·22-s − 1.75·23-s + 0.347·26-s + 0.824·28-s + 0.666·29-s − 0.459·31-s − 0.997·32-s + 0.209·34-s − 0.858·37-s + 0.678·38-s + 0.234·41-s + 0.622·43-s − 1.54·44-s + 0.773·46-s + 1.20·47-s + 0.0502·49-s + 0.631·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(70.2866\)
Root analytic conductor: \(8.38371\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(0.7380388185\)
\(L(\frac12)\) \(\approx\) \(0.7380388185\)
\(L(\frac{9}{2})\) 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 \( 1 + 5 T + p^{7} T^{2} \)
7 \( 1 + 930 T + p^{7} T^{2} \)
11 \( 1 - 8450 T + p^{7} T^{2} \)
13 \( 1 + 6220 T + p^{7} T^{2} \)
17 \( 1 + 9590 T + p^{7} T^{2} \)
19 \( 1 + 45884 T + p^{7} T^{2} \)
23 \( 1 + 4440 p T + p^{7} T^{2} \)
29 \( 1 - 87550 T + p^{7} T^{2} \)
31 \( 1 + 76212 T + p^{7} T^{2} \)
37 \( 1 + 264440 T + p^{7} T^{2} \)
41 \( 1 - 103600 T + p^{7} T^{2} \)
43 \( 1 - 324680 T + p^{7} T^{2} \)
47 \( 1 - 855880 T + p^{7} T^{2} \)
53 \( 1 + 958190 T + p^{7} T^{2} \)
59 \( 1 + 1239550 T + p^{7} T^{2} \)
61 \( 1 - 628522 T + p^{7} T^{2} \)
67 \( 1 + 310380 T + p^{7} T^{2} \)
71 \( 1 - 3934300 T + p^{7} T^{2} \)
73 \( 1 - 4556090 T + p^{7} T^{2} \)
79 \( 1 - 5371644 T + p^{7} T^{2} \)
83 \( 1 + 6711060 T + p^{7} T^{2} \)
89 \( 1 - 3346500 T + p^{7} T^{2} \)
97 \( 1 + 15829730 T + p^{7} T^{2} \)
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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

−10.73353993538823877838355873177, −9.733661888702421367735472831931, −9.188657806569647718208339672327, −8.267778526699003535558187127381, −6.90018106884949522900594907912, −6.07740484898910050461792700199, −4.44137443041451709694477958287, −3.72384734812860435034757589473, −1.95076330941632227941950954174, −0.47414023205130675160685824599, 0.47414023205130675160685824599, 1.95076330941632227941950954174, 3.72384734812860435034757589473, 4.44137443041451709694477958287, 6.07740484898910050461792700199, 6.90018106884949522900594907912, 8.267778526699003535558187127381, 9.188657806569647718208339672327, 9.733661888702421367735472831931, 10.73353993538823877838355873177

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