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Properties

Label 2-35e2-7.5-c0-0-0
Degree $2$
Conductor $1225$
Sign $-0.895 - 0.444i$
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  + (−0.5 + 0.866i)2-s − 8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s + (0.5 − 0.866i)16-s + (−0.499 − 0.866i)18-s − 0.999·22-s + (−0.5 + 0.866i)23-s − 29-s + (−0.5 + 0.866i)37-s + 43-s + (−0.499 − 0.866i)46-s + (1 + 1.73i)53-s + (0.5 − 0.866i)58-s + 64-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s − 8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s + (0.5 − 0.866i)16-s + (−0.499 − 0.866i)18-s − 0.999·22-s + (−0.5 + 0.866i)23-s − 29-s + (−0.5 + 0.866i)37-s + 43-s + (−0.499 − 0.866i)46-s + (1 + 1.73i)53-s + (0.5 − 0.866i)58-s + 64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $-0.895 - 0.444i$
Analytic conductor: \(0.611354\)
Root analytic conductor: \(0.781891\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1225,\ (\ :0),\ -0.895 - 0.444i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6969593240\)
\(L(\frac12)\) \(\approx\) \(0.6969593240\)
\(L(1)\) 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 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
3 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.03313300817045812074929488173, −9.229758580470922955291942117830, −8.576136475883728372990583116844, −7.59842176902218999053938089307, −7.30725607402941585738144896994, −6.19099761555383735231315200215, −5.49525113082812853994718989058, −4.37372423176520755756869262364, −3.15985079605847204754569431114, −1.95256991266311545215048999760, 0.69949591763161655647229985242, 2.10925468469655311819240858735, 3.20266447089468753644929897930, 3.98645908157874496953427827379, 5.57428619416559246494646842947, 6.13763063646913018186982607990, 7.04006346212564303124405887976, 8.372531608678366215526791706188, 8.924255916202679686666584223595, 9.578040823003165920550467263826

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