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Properties

Label 2-775-25.21-c1-0-74
Degree $2$
Conductor $775$
Sign $0.968 - 0.248i$
Analytic cond. $6.18840$
Root an. cond. $2.48765$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.618 − 1.90i)2-s + (−2.61 − 1.90i)3-s + (−1.61 − 1.17i)4-s + (−0.690 − 2.12i)5-s + (−5.23 + 3.80i)6-s − 0.236·7-s + (2.30 + 7.10i)9-s − 4.47·10-s + (−1.19 + 3.66i)11-s + (1.99 + 6.15i)12-s + (0.736 + 2.26i)13-s + (−0.145 + 0.449i)14-s + (−2.23 + 6.88i)15-s + (−1.23 − 3.80i)16-s + (−3.42 + 2.48i)17-s + 14.9·18-s + ⋯
L(s)  = 1  + (0.437 − 1.34i)2-s + (−1.51 − 1.09i)3-s + (−0.809 − 0.587i)4-s + (−0.309 − 0.951i)5-s + (−2.13 + 1.55i)6-s − 0.0892·7-s + (0.769 + 2.36i)9-s − 1.41·10-s + (−0.359 + 1.10i)11-s + (0.577 + 1.77i)12-s + (0.204 + 0.628i)13-s + (−0.0389 + 0.120i)14-s + (−0.577 + 1.77i)15-s + (−0.309 − 0.951i)16-s + (−0.831 + 0.603i)17-s + 3.52·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $0.968 - 0.248i$
Analytic conductor: \(6.18840\)
Root analytic conductor: \(2.48765\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{775} (621, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 775,\ (\ :1/2),\ 0.968 - 0.248i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.690 + 2.12i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
good2 \( 1 + (-0.618 + 1.90i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (2.61 + 1.90i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + 0.236T + 7T^{2} \)
11 \( 1 + (1.19 - 3.66i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-0.736 - 2.26i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (3.42 - 2.48i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (3.61 - 2.62i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-1.16 + 3.57i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (1.80 + 1.31i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 + (0.336 + 1.03i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.336 + 1.03i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 5.47T + 43T^{2} \)
47 \( 1 + (8.85 + 6.43i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-9.35 - 6.79i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.39 + 7.38i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (4.54 - 13.9i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (7.73 - 5.62i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (2.73 + 1.98i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.59 + 4.89i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (6.54 + 4.75i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (3.73 - 2.71i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (1.01 - 3.13i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (8.59 + 6.24i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.987392838520760639279980476220, −8.720353949129594623956637135337, −7.57858368723485416355068929148, −6.77248025784407984166974050702, −5.74752596007813163764537479612, −4.67563187837762314055856812751, −4.22944408150659013634895132643, −2.15035606928965157369944464482, −1.47053224577516000649839244858, 0, 3.23566686881809628221325858995, 4.28693446464228828354710006691, 5.14061269059430734107417553158, 5.92744282816467758175456745830, 6.47110584476055401680976624930, 7.25660558699179520237663377415, 8.389470347769528985902114354666, 9.487807152769531592122830309286, 10.57865723270985186694862354410

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