Properties

Degree $2$
Conductor $1225$
Sign $0.751 - 0.660i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.67 + 0.448i)2-s + (1.73 + 1.00i)4-s + (1.22 + 1.22i)8-s + (−0.866 + 0.5i)9-s + (0.5 − 0.866i)11-s + (0.500 + 0.866i)16-s + (−1.67 + 0.448i)18-s + (1.22 − 1.22i)22-s + (−0.448 + 1.67i)23-s i·29-s − 2·36-s + (−1.67 − 0.448i)37-s + (−1.22 − 1.22i)43-s + (1.73 − 0.999i)44-s + (−1.50 + 2.59i)46-s + ⋯
L(s)  = 1  + (1.67 + 0.448i)2-s + (1.73 + 1.00i)4-s + (1.22 + 1.22i)8-s + (−0.866 + 0.5i)9-s + (0.5 − 0.866i)11-s + (0.500 + 0.866i)16-s + (−1.67 + 0.448i)18-s + (1.22 − 1.22i)22-s + (−0.448 + 1.67i)23-s i·29-s − 2·36-s + (−1.67 − 0.448i)37-s + (−1.22 − 1.22i)43-s + (1.73 − 0.999i)44-s + (−1.50 + 2.59i)46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $0.751 - 0.660i$
Motivic weight: \(0\)
Character: $\chi_{1225} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1225,\ (\ :0),\ 0.751 - 0.660i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.525007766\)
\(L(\frac12)\) \(\approx\) \(2.525007766\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-1.67 - 0.448i)T + (0.866 + 0.5i)T^{2} \)
3 \( 1 + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (-0.866 + 0.5i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.448 - 1.67i)T + (-0.866 - 0.5i)T^{2} \)
29 \( 1 + iT - T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (1.67 + 0.448i)T + (0.866 + 0.5i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
47 \( 1 + (0.866 + 0.5i)T^{2} \)
53 \( 1 + (0.866 - 0.5i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.448 - 1.67i)T + (-0.866 + 0.5i)T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 + (0.866 - 0.5i)T^{2} \)
79 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
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

−10.20975110728442040173566623977, −9.013240639760521944739305929007, −8.149833043429357400684200940703, −7.29786471094650255297087757286, −6.39058873451255708450347278103, −5.61893870590865796451789136215, −5.14345942101640519673494507637, −3.86968123972594648253967970894, −3.31354517513423685009325684271, −2.11991578274121113102865490538, 1.80517201745387408000729403296, 2.89298250954082879827283951574, 3.71606341840407907275218434847, 4.67131232543198701122677113204, 5.33283768543182547513374423079, 6.47308833896276499174317501134, 6.73398714500897558099284955613, 8.174181667813337603160175118819, 9.089357583040083784776658982188, 10.13308979483274934074866268652

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