Properties

Degree $2$
Conductor $1225$
Sign $-0.0560 + 0.998i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.448 − 1.67i)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 + (0.448 − 1.67i)18-s + (1.22 − 1.22i)22-s + (1.67 − 0.448i)23-s i·29-s − 2·36-s + (0.448 + 1.67i)37-s + (−1.22 − 1.22i)43-s + (−1.73 − 0.999i)44-s + (−1.50 − 2.59i)46-s + ⋯
L(s)  = 1  + (−0.448 − 1.67i)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 + (0.448 − 1.67i)18-s + (1.22 − 1.22i)22-s + (1.67 − 0.448i)23-s i·29-s − 2·36-s + (0.448 + 1.67i)37-s + (−1.22 − 1.22i)43-s + (−1.73 − 0.999i)44-s + (−1.50 − 2.59i)46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0560 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0560 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $-0.0560 + 0.998i$
Motivic weight: \(0\)
Character: $\chi_{1225} (618, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1225,\ (\ :0),\ -0.0560 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8436642980\)
\(L(\frac12)\) \(\approx\) \(0.8436642980\)
\(L(1)\) 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 \( 1 + (0.448 + 1.67i)T + (-0.866 + 0.5i)T^{2} \)
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 \( 1 + (-1.67 + 0.448i)T + (0.866 - 0.5i)T^{2} \)
29 \( 1 + iT - T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.448 - 1.67i)T + (-0.866 + 0.5i)T^{2} \)
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 \( 1 + (1.67 + 0.448i)T + (0.866 + 0.5i)T^{2} \)
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

−9.967680747148809099869578146182, −9.204429242419644365712384849681, −8.433438477190557035997546775908, −7.43843457739352812460050226026, −6.55722367897966169945378338686, −4.91242892508836129973451076878, −4.35851928887375527034853727680, −3.28801632691664035382696438772, −2.22007634335971687663320888564, −1.27619977569278877705028896279, 1.14208435777253230841162635923, 3.29867726749598220788928904031, 4.43791143563083636818897806029, 5.34868943885404227387294991713, 6.18902869469411294274541989396, 6.93184655847768365828126837481, 7.46369628073630797928731723072, 8.479731086286632419181445426448, 9.106040958692596411519265774213, 9.653983699196825994792139047474

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