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Properties

Label 2-775-775.61-c0-0-2
Degree $2$
Conductor $775$
Sign $-0.187 + 0.982i$
Analytic cond. $0.386775$
Root an. cond. $0.621912$
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  + (1.30 − 0.951i)2-s + (0.500 − 1.53i)4-s + (−0.809 − 0.587i)5-s + 0.618·7-s + (−0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s − 1.61·10-s + (0.809 − 0.587i)14-s − 1.61·18-s + (0.190 + 0.587i)19-s + (−1.30 + 0.951i)20-s + (0.309 + 0.951i)25-s + (0.309 − 0.951i)28-s + (0.309 + 0.951i)31-s + 0.999·32-s + ⋯
L(s)  = 1  + (1.30 − 0.951i)2-s + (0.500 − 1.53i)4-s + (−0.809 − 0.587i)5-s + 0.618·7-s + (−0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s − 1.61·10-s + (0.809 − 0.587i)14-s − 1.61·18-s + (0.190 + 0.587i)19-s + (−1.30 + 0.951i)20-s + (0.309 + 0.951i)25-s + (0.309 − 0.951i)28-s + (0.309 + 0.951i)31-s + 0.999·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $-0.187 + 0.982i$
Analytic conductor: \(0.386775\)
Root analytic conductor: \(0.621912\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{775} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 775,\ (\ :0),\ -0.187 + 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.681559111\)
\(L(\frac12)\) \(\approx\) \(1.681559111\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.809 + 0.587i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
3 \( 1 + (0.809 + 0.587i)T^{2} \)
7 \( 1 - 0.618T + T^{2} \)
11 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (-0.309 - 0.951i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + (-0.309 + 0.951i)T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.309 - 0.951i)T^{2} \)
41 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
53 \( 1 + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.309 + 0.951i)T^{2} \)
67 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
71 \( 1 + (-0.618 + 1.90i)T + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.809 - 0.587i)T^{2} \)
89 \( 1 + (-0.309 + 0.951i)T^{2} \)
97 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
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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

−10.80758016980508728834464718917, −9.591351218530067724787291943694, −8.544801889151305788923861148057, −7.83560639508311748537649565799, −6.43703942607549068384431573818, −5.40526509856336253553676443423, −4.71909648391123343508513486815, −3.77529144025918400219114578536, −2.96915284520705345769319498153, −1.45820259384427566239885733439, 2.59646677040222676282941517460, 3.61419537694926660339914449926, 4.60532165955001197682012470155, 5.32852086698375049938421213693, 6.32385843868541750037774545088, 7.17024409315532894496199105191, 7.918892114642332827121208358308, 8.534911891700038551454601559062, 10.04087809987657910680615786351, 11.21493175051225325844322229293

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