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Properties

Label 2-775-5.4-c1-0-33
Degree $2$
Conductor $775$
Sign $0.447 + 0.894i$
Analytic cond. $6.18840$
Root an. cond. $2.48765$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·3-s + 2·4-s + 2·9-s − 4·11-s − 2i·12-s − 6i·13-s + 4·16-s − 5i·17-s + 19-s + 8i·23-s − 5i·27-s + 10·29-s − 31-s + 4i·33-s + 4·36-s − i·37-s + ⋯
L(s)  = 1  − 0.577i·3-s + 4-s + 0.666·9-s − 1.20·11-s − 0.577i·12-s − 1.66i·13-s + 16-s − 1.21i·17-s + 0.229·19-s + 1.66i·23-s − 0.962i·27-s + 1.85·29-s − 0.179·31-s + 0.696i·33-s + 0.666·36-s − 0.164i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(6.18840\)
Root analytic conductor: \(2.48765\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{775} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 775,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64554 - 1.01700i\)
\(L(\frac12)\) \(\approx\) \(1.64554 - 1.01700i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
31 \( 1 + T \)
good2 \( 1 - 2T^{2} \)
3 \( 1 + iT - 3T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + 6iT - 13T^{2} \)
17 \( 1 + 5iT - 17T^{2} \)
19 \( 1 - T + 19T^{2} \)
23 \( 1 - 8iT - 23T^{2} \)
29 \( 1 - 10T + 29T^{2} \)
37 \( 1 + iT - 37T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 + 7iT - 43T^{2} \)
47 \( 1 - 6iT - 47T^{2} \)
53 \( 1 - 5iT - 53T^{2} \)
59 \( 1 + 11T + 59T^{2} \)
61 \( 1 + 12T + 61T^{2} \)
67 \( 1 - 2iT - 67T^{2} \)
71 \( 1 - 9T + 71T^{2} \)
73 \( 1 + 9iT - 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 - 9iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 14iT - 97T^{2} \)
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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

−10.41044303706651300394071596226, −9.466003611325508049518057608641, −7.890394711819109950625215206864, −7.70915822697217151511001502685, −6.84439527914552988698813881367, −5.76554771935643504569960634164, −5.00300429306524236642947842196, −3.25378305761747039797499486378, −2.48207714564487906477891693061, −1.03228321438965306959745188242, 1.70655469139794557059481133399, 2.81589308501884073096171315419, 4.12339276751089641210516377440, 4.94007675554633247580204355813, 6.28466943321559373227459233947, 6.83214982711235958026752157333, 7.88632026386161103566314793055, 8.722197186877822478329589311283, 9.904256373478041683973361706975, 10.48278345533446524992325430119

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