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Properties

Label 2-15e2-15.2-c5-0-17
Degree $2$
Conductor $225$
Sign $0.999 - 0.0387i$
Analytic cond. $36.0863$
Root an. cond. $6.00719$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.923 − 0.923i)2-s + 30.2i·4-s + (−31.9 − 31.9i)7-s + (57.5 + 57.5i)8-s − 375. i·11-s + (278. − 278. i)13-s − 59.0·14-s − 863.·16-s + (812. − 812. i)17-s − 23.2i·19-s + (−347. − 347. i)22-s + (1.76e3 + 1.76e3i)23-s − 514. i·26-s + (968. − 968. i)28-s + 4.94e3·29-s + ⋯
L(s)  = 1  + (0.163 − 0.163i)2-s + 0.946i·4-s + (−0.246 − 0.246i)7-s + (0.317 + 0.317i)8-s − 0.936i·11-s + (0.456 − 0.456i)13-s − 0.0805·14-s − 0.842·16-s + (0.681 − 0.681i)17-s − 0.0147i·19-s + (−0.152 − 0.152i)22-s + (0.694 + 0.694i)23-s − 0.149i·26-s + (0.233 − 0.233i)28-s + 1.09·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0387i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0387i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.999 - 0.0387i$
Analytic conductor: \(36.0863\)
Root analytic conductor: \(6.00719\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :5/2),\ 0.999 - 0.0387i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.219088542\)
\(L(\frac12)\) \(\approx\) \(2.219088542\)
\(L(\frac{7}{2})\) 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 \( 1 + (-0.923 + 0.923i)T - 32iT^{2} \)
7 \( 1 + (31.9 + 31.9i)T + 1.68e4iT^{2} \)
11 \( 1 + 375. iT - 1.61e5T^{2} \)
13 \( 1 + (-278. + 278. i)T - 3.71e5iT^{2} \)
17 \( 1 + (-812. + 812. i)T - 1.41e6iT^{2} \)
19 \( 1 + 23.2iT - 2.47e6T^{2} \)
23 \( 1 + (-1.76e3 - 1.76e3i)T + 6.43e6iT^{2} \)
29 \( 1 - 4.94e3T + 2.05e7T^{2} \)
31 \( 1 + 292.T + 2.86e7T^{2} \)
37 \( 1 + (-8.12e3 - 8.12e3i)T + 6.93e7iT^{2} \)
41 \( 1 - 5.09e3iT - 1.15e8T^{2} \)
43 \( 1 + (-7.77e3 + 7.77e3i)T - 1.47e8iT^{2} \)
47 \( 1 + (1.55e4 - 1.55e4i)T - 2.29e8iT^{2} \)
53 \( 1 + (-2.62e4 - 2.62e4i)T + 4.18e8iT^{2} \)
59 \( 1 - 4.29e4T + 7.14e8T^{2} \)
61 \( 1 + 2.07e4T + 8.44e8T^{2} \)
67 \( 1 + (3.97e3 + 3.97e3i)T + 1.35e9iT^{2} \)
71 \( 1 + 5.09e4iT - 1.80e9T^{2} \)
73 \( 1 + (-5.69e4 + 5.69e4i)T - 2.07e9iT^{2} \)
79 \( 1 + 6.90e4iT - 3.07e9T^{2} \)
83 \( 1 + (-7.36e4 - 7.36e4i)T + 3.93e9iT^{2} \)
89 \( 1 - 5.39e4T + 5.58e9T^{2} \)
97 \( 1 + (-2.52e4 - 2.52e4i)T + 8.58e9iT^{2} \)
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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

−11.49308604040128628828105678122, −10.59817693133106480861001758793, −9.338667942600491218560723458689, −8.321637966926893801726199144441, −7.51082051346229237271144056369, −6.30860513195607260212001896856, −4.97410132803776846529214235993, −3.60342965553947148839971122745, −2.82719643242123047681556455294, −0.857288603918323752613495073919, 0.946249043427315851287857697050, 2.27647581114958426046718856559, 4.05378641402363125829894931837, 5.16973581062435253813833050171, 6.23349005686425944303568616752, 7.08754777473340445263791786272, 8.500748416573200814343439527809, 9.579102307451593824905365912055, 10.29891137864840783600504624806, 11.27433465691920334113801906138

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