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Properties

Label 2-775-31.28-c1-0-31
Degree $2$
Conductor $775$
Sign $0.999 - 0.0301i$
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  + (1.78 − 1.29i)2-s + (−0.276 + 2.63i)3-s + (0.887 − 2.73i)4-s + (2.91 + 5.05i)6-s + (3.81 − 0.809i)7-s + (−0.594 − 1.82i)8-s + (−3.90 − 0.830i)9-s + (−1.41 − 1.57i)11-s + (6.93 + 3.08i)12-s + (4.93 − 2.19i)13-s + (5.75 − 6.38i)14-s + (1.21 + 0.880i)16-s + (−4.96 + 5.51i)17-s + (−8.05 + 3.58i)18-s + (−1.56 − 0.694i)19-s + ⋯
L(s)  = 1  + (1.26 − 0.917i)2-s + (−0.159 + 1.51i)3-s + (0.443 − 1.36i)4-s + (1.19 + 2.06i)6-s + (1.44 − 0.306i)7-s + (−0.210 − 0.646i)8-s + (−1.30 − 0.276i)9-s + (−0.426 − 0.473i)11-s + (2.00 + 0.891i)12-s + (1.36 − 0.609i)13-s + (1.53 − 1.70i)14-s + (0.303 + 0.220i)16-s + (−1.20 + 1.33i)17-s + (−1.89 + 0.845i)18-s + (−0.358 − 0.159i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0301i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0301i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $0.999 - 0.0301i$
Analytic conductor: \(6.18840\)
Root analytic conductor: \(2.48765\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{775} (276, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 775,\ (\ :1/2),\ 0.999 - 0.0301i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.25904 + 0.0490687i\)
\(L(\frac12)\) \(\approx\) \(3.25904 + 0.0490687i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.64 + 4.90i)T \)
good2 \( 1 + (-1.78 + 1.29i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.276 - 2.63i)T + (-2.93 - 0.623i)T^{2} \)
7 \( 1 + (-3.81 + 0.809i)T + (6.39 - 2.84i)T^{2} \)
11 \( 1 + (1.41 + 1.57i)T + (-1.14 + 10.9i)T^{2} \)
13 \( 1 + (-4.93 + 2.19i)T + (8.69 - 9.66i)T^{2} \)
17 \( 1 + (4.96 - 5.51i)T + (-1.77 - 16.9i)T^{2} \)
19 \( 1 + (1.56 + 0.694i)T + (12.7 + 14.1i)T^{2} \)
23 \( 1 + (-0.672 - 2.06i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (0.124 - 0.0904i)T + (8.96 - 27.5i)T^{2} \)
37 \( 1 + (-0.0473 - 0.0820i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.302 - 2.87i)T + (-40.1 + 8.52i)T^{2} \)
43 \( 1 + (-1.79 - 0.797i)T + (28.7 + 31.9i)T^{2} \)
47 \( 1 + (2.66 + 1.93i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (13.5 + 2.88i)T + (48.4 + 21.5i)T^{2} \)
59 \( 1 + (-1.10 + 10.4i)T + (-57.7 - 12.2i)T^{2} \)
61 \( 1 - 8.90T + 61T^{2} \)
67 \( 1 + (3.32 - 5.75i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (12.8 + 2.73i)T + (64.8 + 28.8i)T^{2} \)
73 \( 1 + (4.65 + 5.16i)T + (-7.63 + 72.6i)T^{2} \)
79 \( 1 + (7.81 - 8.67i)T + (-8.25 - 78.5i)T^{2} \)
83 \( 1 + (-0.413 - 3.93i)T + (-81.1 + 17.2i)T^{2} \)
89 \( 1 + (-0.968 + 2.98i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-3.94 + 12.1i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.80664214531752784754642188748, −9.970637990311934379966049968875, −8.607447527839613625716092530241, −8.126815267749067846207148080926, −6.16001765412573806166887261781, −5.42858941645932697315152870352, −4.53450328121371052518143757794, −4.08070781502262186679300407593, −3.15735409421998309425247986011, −1.69239184213496549442768611769, 1.45132817381271321880748948112, 2.62034176858844472686203368322, 4.32507232074989624628229923282, 5.02367558241470265461713489824, 6.02883107933561710221395617566, 6.73438481527926519561544358606, 7.39738324170219406839737511465, 8.171344107793094261151271967959, 8.925725792766358123757398963139, 10.74966581389651964558542141828

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