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Properties

Label 2-15e2-5.2-c4-0-4
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: $\chi_{225} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :2),\ -0.525 + 0.850i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.154851927\)
\(L(\frac12)\) \(\approx\) \(1.154851927\)
\(L(3)\) 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

−12.55421471218701869351066010068, −11.69213355904975717987345693736, −10.23430204421217441767462945683, −9.119588414941252156007962277407, −7.60390544468379299004763625416, −7.10114394777268596358748759672, −6.13959574203618738266371137518, −4.89533719470764430401358474290, −4.08933322307680087666755610289, −2.72800090046431820289679764458, 0.23231090340722660852265594036, 2.31446904081383294403819208594, 2.92995721537934280711402688583, 4.35928677832489144257749067065, 5.43413550648884262308046698042, 6.26406391894852383770534352395, 7.982799949614261600375029719650, 9.487072320517781640050936444603, 10.26974151151785631110932435666, 11.06389114268157560372897216401

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