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Properties

Label 2-15e2-5.2-c0-0-0
Degree $2$
Conductor $225$
Sign $0.850 + 0.525i$
Analytic cond. $0.112289$
Root an. cond. $0.335096$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·4-s − 16-s + 2i·19-s − 2·31-s − i·49-s + 2·61-s + i·64-s + 2·76-s − 2i·79-s − 2i·109-s + ⋯
L(s)  = 1  − i·4-s − 16-s + 2i·19-s − 2·31-s − i·49-s + 2·61-s + i·64-s + 2·76-s − 2i·79-s − 2i·109-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.850 + 0.525i$
Analytic conductor: \(0.112289\)
Root analytic conductor: \(0.335096\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :0),\ 0.850 + 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7108859674\)
\(L(\frac12)\) \(\approx\) \(0.7108859674\)
\(L(1)\) 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 \( 1 + iT^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 - 2iT - T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + 2T + T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 2T + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + 2iT - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + iT^{2} \)
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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

−12.37548083065245157988382609152, −11.31462212756581432986790872170, −10.39128633790638138643671563795, −9.662959581269237668744357861175, −8.558663126224290290571978918038, −7.30990867419776323575653941912, −6.09326950367494740805627823014, −5.26331522599767078774420425881, −3.79449030796393505024174188555, −1.82341514415177521088400579334, 2.57334003132524961343902120842, 3.88505578156172457214518408787, 5.13878211294140034503208333172, 6.73232390186078945006518717104, 7.53069093482151041493926454554, 8.679914944651945921251544356863, 9.419581453182552381230800138331, 10.90829519439838618163105865185, 11.56606673163933529207753982522, 12.69803762574422672434967233354

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