Properties

Label 2-15e2-1.1-c5-0-3
Degree $2$
Conductor $225$
Sign $1$
Analytic cond. $36.0863$
Root an. cond. $6.00719$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 28·4-s − 192·7-s − 120·8-s + 148·11-s − 286·13-s − 384·14-s + 656·16-s − 1.67e3·17-s + 1.06e3·19-s + 296·22-s + 2.97e3·23-s − 572·26-s + 5.37e3·28-s + 3.41e3·29-s − 2.44e3·31-s + 5.15e3·32-s − 3.35e3·34-s − 182·37-s + 2.12e3·38-s + 9.39e3·41-s + 1.24e3·43-s − 4.14e3·44-s + 5.95e3·46-s − 1.20e4·47-s + 2.00e4·49-s + 8.00e3·52-s + ⋯
L(s)  = 1  + 0.353·2-s − 7/8·4-s − 1.48·7-s − 0.662·8-s + 0.368·11-s − 0.469·13-s − 0.523·14-s + 0.640·16-s − 1.40·17-s + 0.673·19-s + 0.130·22-s + 1.17·23-s − 0.165·26-s + 1.29·28-s + 0.752·29-s − 0.457·31-s + 0.889·32-s − 0.497·34-s − 0.0218·37-s + 0.238·38-s + 0.873·41-s + 0.102·43-s − 0.322·44-s + 0.414·46-s − 0.798·47-s + 1.19·49-s + 0.410·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(36.0863\)
Root analytic conductor: \(6.00719\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: $\chi_{225} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(1.125854998\)
\(L(\frac12)\) \(\approx\) \(1.125854998\)
\(L(\frac{7}{2})\) 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 \( 1 - p T + p^{5} T^{2} \)
7 \( 1 + 192 T + p^{5} T^{2} \)
11 \( 1 - 148 T + p^{5} T^{2} \)
13 \( 1 + 22 p T + p^{5} T^{2} \)
17 \( 1 + 1678 T + p^{5} T^{2} \)
19 \( 1 - 1060 T + p^{5} T^{2} \)
23 \( 1 - 2976 T + p^{5} T^{2} \)
29 \( 1 - 3410 T + p^{5} T^{2} \)
31 \( 1 + 2448 T + p^{5} T^{2} \)
37 \( 1 + 182 T + p^{5} T^{2} \)
41 \( 1 - 9398 T + p^{5} T^{2} \)
43 \( 1 - 1244 T + p^{5} T^{2} \)
47 \( 1 + 12088 T + p^{5} T^{2} \)
53 \( 1 - 23846 T + p^{5} T^{2} \)
59 \( 1 - 20020 T + p^{5} T^{2} \)
61 \( 1 - 32302 T + p^{5} T^{2} \)
67 \( 1 + 60972 T + p^{5} T^{2} \)
71 \( 1 - 32648 T + p^{5} T^{2} \)
73 \( 1 - 38774 T + p^{5} T^{2} \)
79 \( 1 + 33360 T + p^{5} T^{2} \)
83 \( 1 - 16716 T + p^{5} T^{2} \)
89 \( 1 + 101370 T + p^{5} T^{2} \)
97 \( 1 - 119038 T + p^{5} T^{2} \)
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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

−11.53443069792532219571444385364, −10.19369516411019024976740967763, −9.362619038709084687147553466051, −8.735802220224539128034261988738, −7.14066858548824096262243770559, −6.21870557482653431446601375533, −5.00126213651318958829567681443, −3.86132829810621868924906174712, −2.79039576886860897226069523278, −0.59015544588329404356663833953, 0.59015544588329404356663833953, 2.79039576886860897226069523278, 3.86132829810621868924906174712, 5.00126213651318958829567681443, 6.21870557482653431446601375533, 7.14066858548824096262243770559, 8.735802220224539128034261988738, 9.362619038709084687147553466051, 10.19369516411019024976740967763, 11.53443069792532219571444385364

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