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Properties

Label 2-775-5.4-c1-0-21
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  + 2.80i·2-s − 0.342i·3-s − 5.84·4-s + 0.958·6-s − 1.04i·7-s − 10.7i·8-s + 2.88·9-s + 4.64·11-s + 1.99i·12-s + 2.95i·13-s + 2.91·14-s + 18.4·16-s − 6.29i·17-s + 8.07i·18-s + 1.11·19-s + ⋯
L(s)  = 1  + 1.98i·2-s − 0.197i·3-s − 2.92·4-s + 0.391·6-s − 0.393i·7-s − 3.80i·8-s + 0.960·9-s + 1.39·11-s + 0.577i·12-s + 0.820i·13-s + 0.779·14-s + 4.61·16-s − 1.52i·17-s + 1.90i·18-s + 0.255·19-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

\(L(1)\) \(\approx\) \(0.747149 + 1.20891i\)
\(L(\frac12)\) \(\approx\) \(0.747149 + 1.20891i\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.80iT - 2T^{2} \)
3 \( 1 + 0.342iT - 3T^{2} \)
7 \( 1 + 1.04iT - 7T^{2} \)
11 \( 1 - 4.64T + 11T^{2} \)
13 \( 1 - 2.95iT - 13T^{2} \)
17 \( 1 + 6.29iT - 17T^{2} \)
19 \( 1 - 1.11T + 19T^{2} \)
23 \( 1 - 3.87iT - 23T^{2} \)
29 \( 1 + 2.35T + 29T^{2} \)
37 \( 1 + 1.30iT - 37T^{2} \)
41 \( 1 - 1.92T + 41T^{2} \)
43 \( 1 + 4.29iT - 43T^{2} \)
47 \( 1 - 3.31iT - 47T^{2} \)
53 \( 1 - 5.10iT - 53T^{2} \)
59 \( 1 - 5.07T + 59T^{2} \)
61 \( 1 - 5.31T + 61T^{2} \)
67 \( 1 - 8.64iT - 67T^{2} \)
71 \( 1 + 3.88T + 71T^{2} \)
73 \( 1 - 12.1iT - 73T^{2} \)
79 \( 1 - 7.12T + 79T^{2} \)
83 \( 1 + 7.46iT - 83T^{2} \)
89 \( 1 - 12.1T + 89T^{2} \)
97 \( 1 - 0.915iT - 97T^{2} \)
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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

−9.997052704350479239289622256146, −9.382727553729489960264933975302, −8.828433759217664263772409208641, −7.53734967944177114371662173348, −7.13315103901710295249406496762, −6.52350500252799068999247630399, −5.45944289591731970550682417299, −4.42617730970738440312616061308, −3.81743286109421867117071021930, −1.06367452552188735021578541289, 1.10588626570455372779707229975, 2.13465406434001909532194535309, 3.51343989585090309484432783875, 4.08190760865808895347378086326, 5.08274354888415721266557506433, 6.27327624067478177706809175228, 7.926473730528597651535732987814, 8.781006125651909763765900127197, 9.463852784204710052397076639336, 10.24754574510456826233061667018

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