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Properties

Label 2-35e2-35.18-c0-0-1
Degree $2$
Conductor $1225$
Sign $0.496 + 0.867i$
Analytic cond. $0.611354$
Root an. cond. $0.781891$
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  + (−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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.496 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.496 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $0.496 + 0.867i$
Analytic conductor: \(0.611354\)
Root analytic conductor: \(0.781891\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1225,\ (\ :0),\ 0.496 + 0.867i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4493610666\)
\(L(\frac12)\) \(\approx\) \(0.4493610666\)
\(L(1)\) 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

−9.602043827071134545731981088649, −9.045220977357553702149470134919, −8.207813644904337253833370774792, −7.87476645177958270627845876133, −6.63312760434713522313148700277, −5.95275891675322451766954676615, −4.53426284451375565156702819608, −3.06351547788081039922688455579, −2.31278192804378191283822101731, −0.795073279066333925922302132925, 1.19578509012443993254680863876, 2.46475113049638324653093542350, 3.84273784607651024876426960417, 5.35309891183750941152610222615, 6.20158632784032086414084765505, 7.08478272532707150749643069449, 7.63137124690433274908751214152, 8.591689426133857874931771599756, 9.278278982576833044602296419632, 9.642920884418739934945132279550

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