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Properties

Label 2-15e2-5.3-c4-0-8
Degree $2$
Conductor $225$
Sign $-0.525 - 0.850i$
Analytic cond. $23.2582$
Root an. cond. $4.82267$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5 + 5i)2-s − 34i·4-s + (−40 + 40i)7-s + (90 + 90i)8-s + 100·11-s + (−205 − 205i)13-s − 400i·14-s − 356·16-s + (235 − 235i)17-s − 72i·19-s + (−500 + 500i)22-s + (340 + 340i)23-s + 2.05e3·26-s + (1.36e3 + 1.36e3i)28-s + 450i·29-s + ⋯
L(s)  = 1  + (−1.25 + 1.25i)2-s − 2.12i·4-s + (−0.816 + 0.816i)7-s + (1.40 + 1.40i)8-s + 0.826·11-s + (−1.21 − 1.21i)13-s − 2.04i·14-s − 1.39·16-s + (0.813 − 0.813i)17-s − 0.199i·19-s + (−1.03 + 1.03i)22-s + (0.642 + 0.642i)23-s + 3.03·26-s + (1.73 + 1.73i)28-s + 0.535i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.525 - 0.850i$
Analytic conductor: \(23.2582\)
Root analytic conductor: \(4.82267\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :2),\ -0.525 - 0.850i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.6961814615\)
\(L(\frac12)\) \(\approx\) \(0.6961814615\)
\(L(3)\) 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 \( 1 + (5 - 5i)T - 16iT^{2} \)
7 \( 1 + (40 - 40i)T - 2.40e3iT^{2} \)
11 \( 1 - 100T + 1.46e4T^{2} \)
13 \( 1 + (205 + 205i)T + 2.85e4iT^{2} \)
17 \( 1 + (-235 + 235i)T - 8.35e4iT^{2} \)
19 \( 1 + 72iT - 1.30e5T^{2} \)
23 \( 1 + (-340 - 340i)T + 2.79e5iT^{2} \)
29 \( 1 - 450iT - 7.07e5T^{2} \)
31 \( 1 - 428T + 9.23e5T^{2} \)
37 \( 1 + (-755 + 755i)T - 1.87e6iT^{2} \)
41 \( 1 + 950T + 2.82e6T^{2} \)
43 \( 1 + (-1.22e3 - 1.22e3i)T + 3.41e6iT^{2} \)
47 \( 1 + (320 - 320i)T - 4.87e6iT^{2} \)
53 \( 1 + (-505 - 505i)T + 7.89e6iT^{2} \)
59 \( 1 - 6.30e3iT - 1.21e7T^{2} \)
61 \( 1 + 3.80e3T + 1.38e7T^{2} \)
67 \( 1 + (340 - 340i)T - 2.01e7iT^{2} \)
71 \( 1 - 3.40e3T + 2.54e7T^{2} \)
73 \( 1 + (415 + 415i)T + 2.83e7iT^{2} \)
79 \( 1 - 6.73e3iT - 3.89e7T^{2} \)
83 \( 1 + (680 + 680i)T + 4.74e7iT^{2} \)
89 \( 1 - 2.25e3iT - 6.27e7T^{2} \)
97 \( 1 + (1.61e3 - 1.61e3i)T - 8.85e7iT^{2} \)
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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

−11.85551565989374780469247076040, −10.41952192063191232877441505612, −9.530864410124089713032656812577, −9.082939296319899880194501579623, −7.83279676325583435813297537143, −7.08091145760234816016596989323, −6.00453033174155040040550302402, −5.16558009206864449825766281670, −2.90560698767633612603061888739, −0.862739879971723463749068713037, 0.51567598124899780364380947199, 1.84086838659847668743561398281, 3.25326031773844633564459783131, 4.30979887746477267789746802694, 6.54134463421956447481481867721, 7.48053821921976246047605180182, 8.646641004581178818301940321185, 9.663097263903648864458703601541, 10.04268257726611438736544189119, 11.11385556199981111731292423258

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