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Properties

Label 2-775-155.59-c1-0-11
Degree $2$
Conductor $775$
Sign $0.948 - 0.315i$
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  + (−0.0622 − 0.0856i)2-s + (−0.541 − 0.0569i)3-s + (0.614 − 1.89i)4-s + (0.0288 + 0.0499i)6-s + (0.520 + 2.44i)7-s + (−0.401 + 0.130i)8-s + (−2.64 − 0.561i)9-s + (2.63 + 2.92i)11-s + (−0.440 + 0.990i)12-s + (1.62 + 3.65i)13-s + (0.177 − 0.196i)14-s + (−3.18 − 2.31i)16-s + (3.49 + 3.14i)17-s + (0.116 + 0.261i)18-s + (3.29 + 1.46i)19-s + ⋯
L(s)  = 1  + (−0.0440 − 0.0605i)2-s + (−0.312 − 0.0328i)3-s + (0.307 − 0.945i)4-s + (0.0117 + 0.0203i)6-s + (0.196 + 0.924i)7-s + (−0.141 + 0.0461i)8-s + (−0.881 − 0.187i)9-s + (0.793 + 0.881i)11-s + (−0.127 + 0.285i)12-s + (0.451 + 1.01i)13-s + (0.0473 − 0.0525i)14-s + (−0.795 − 0.577i)16-s + (0.847 + 0.763i)17-s + (0.0274 + 0.0616i)18-s + (0.756 + 0.336i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.315i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 - 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $0.948 - 0.315i$
Analytic conductor: \(6.18840\)
Root analytic conductor: \(2.48765\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{775} (524, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 775,\ (\ :1/2),\ 0.948 - 0.315i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41498 + 0.229336i\)
\(L(\frac12)\) \(\approx\) \(1.41498 + 0.229336i\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.37 - 1.43i)T \)
good2 \( 1 + (0.0622 + 0.0856i)T + (-0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.541 + 0.0569i)T + (2.93 + 0.623i)T^{2} \)
7 \( 1 + (-0.520 - 2.44i)T + (-6.39 + 2.84i)T^{2} \)
11 \( 1 + (-2.63 - 2.92i)T + (-1.14 + 10.9i)T^{2} \)
13 \( 1 + (-1.62 - 3.65i)T + (-8.69 + 9.66i)T^{2} \)
17 \( 1 + (-3.49 - 3.14i)T + (1.77 + 16.9i)T^{2} \)
19 \( 1 + (-3.29 - 1.46i)T + (12.7 + 14.1i)T^{2} \)
23 \( 1 + (1.62 - 0.529i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (-6.40 + 4.65i)T + (8.96 - 27.5i)T^{2} \)
37 \( 1 + (-7.51 + 4.33i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.449 + 4.27i)T + (-40.1 + 8.52i)T^{2} \)
43 \( 1 + (-2.00 + 4.50i)T + (-28.7 - 31.9i)T^{2} \)
47 \( 1 + (4.35 - 5.99i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.25 - 5.91i)T + (-48.4 - 21.5i)T^{2} \)
59 \( 1 + (0.494 - 4.70i)T + (-57.7 - 12.2i)T^{2} \)
61 \( 1 - 0.682T + 61T^{2} \)
67 \( 1 + (6.77 + 3.91i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-13.9 - 2.96i)T + (64.8 + 28.8i)T^{2} \)
73 \( 1 + (9.08 - 8.17i)T + (7.63 - 72.6i)T^{2} \)
79 \( 1 + (-6.00 + 6.66i)T + (-8.25 - 78.5i)T^{2} \)
83 \( 1 + (-9.98 + 1.04i)T + (81.1 - 17.2i)T^{2} \)
89 \( 1 + (2.24 - 6.90i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-6.65 - 2.16i)T + (78.4 + 57.0i)T^{2} \)
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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

−10.37614825480245886131880919205, −9.405359905988524764419780219976, −8.968229533579732070510258716659, −7.76472684887344102293668756071, −6.54791695952271246070165179496, −5.96793112364307878771128157444, −5.23578477349173505840231309394, −3.99651910824184944193634642245, −2.44850563684403483495971725710, −1.36218440966921579973673316045, 0.874463221124599297286518053326, 2.96867121194873273225108582339, 3.53719723941188772826025685194, 4.86391125880356822715355160412, 5.94142224749955915083766150775, 6.82769336778449359662388001493, 7.83365700984090684703242601265, 8.309935103116472635267373040615, 9.333259699977243145208941856748, 10.45928532398319642801943429192

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