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Properties

Label 2-775-155.144-c1-0-44
Degree $2$
Conductor $775$
Sign $0.159 + 0.987i$
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  + (2.55 − 0.831i)2-s + (1.05 − 0.949i)3-s + (4.23 − 3.07i)4-s + (1.90 − 3.30i)6-s + (−1.71 + 0.180i)7-s + (5.11 − 7.04i)8-s + (−0.103 + 0.980i)9-s + (−1.22 − 0.543i)11-s + (1.54 − 7.26i)12-s + (0.763 + 3.59i)13-s + (−4.24 + 1.88i)14-s + (4.00 − 12.3i)16-s + (−1.12 − 2.52i)17-s + (0.551 + 2.59i)18-s + (−2.51 − 0.533i)19-s + ⋯
L(s)  = 1  + (1.80 − 0.587i)2-s + (0.608 − 0.548i)3-s + (2.11 − 1.53i)4-s + (0.779 − 1.34i)6-s + (−0.649 + 0.0682i)7-s + (1.80 − 2.49i)8-s + (−0.0343 + 0.326i)9-s + (−0.368 − 0.164i)11-s + (0.446 − 2.09i)12-s + (0.211 + 0.995i)13-s + (−1.13 + 0.504i)14-s + (1.00 − 3.07i)16-s + (−0.272 − 0.612i)17-s + (0.129 + 0.611i)18-s + (−0.576 − 0.122i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.159 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.159 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $0.159 + 0.987i$
Analytic conductor: \(6.18840\)
Root analytic conductor: \(2.48765\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{775} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 775,\ (\ :1/2),\ 0.159 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.76032 - 3.20085i\)
\(L(\frac12)\) \(\approx\) \(3.76032 - 3.20085i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.75 - 2.90i)T \)
good2 \( 1 + (-2.55 + 0.831i)T + (1.61 - 1.17i)T^{2} \)
3 \( 1 + (-1.05 + 0.949i)T + (0.313 - 2.98i)T^{2} \)
7 \( 1 + (1.71 - 0.180i)T + (6.84 - 1.45i)T^{2} \)
11 \( 1 + (1.22 + 0.543i)T + (7.36 + 8.17i)T^{2} \)
13 \( 1 + (-0.763 - 3.59i)T + (-11.8 + 5.28i)T^{2} \)
17 \( 1 + (1.12 + 2.52i)T + (-11.3 + 12.6i)T^{2} \)
19 \( 1 + (2.51 + 0.533i)T + (17.3 + 7.72i)T^{2} \)
23 \( 1 + (-0.316 + 0.436i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (-2.51 - 7.73i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 + (6.70 + 3.87i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.0696 - 0.0773i)T + (-4.28 - 40.7i)T^{2} \)
43 \( 1 + (-0.624 + 2.93i)T + (-39.2 - 17.4i)T^{2} \)
47 \( 1 + (6.39 + 2.07i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-2.78 - 0.292i)T + (51.8 + 11.0i)T^{2} \)
59 \( 1 + (0.311 + 0.346i)T + (-6.16 + 58.6i)T^{2} \)
61 \( 1 + 5.11T + 61T^{2} \)
67 \( 1 + (-7.18 + 4.14i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.497 - 4.73i)T + (-69.4 - 14.7i)T^{2} \)
73 \( 1 + (-3.05 + 6.85i)T + (-48.8 - 54.2i)T^{2} \)
79 \( 1 + (-8.86 + 3.94i)T + (52.8 - 58.7i)T^{2} \)
83 \( 1 + (-12.3 - 11.1i)T + (8.67 + 82.5i)T^{2} \)
89 \( 1 + (12.3 - 9.00i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-0.751 - 1.03i)T + (-29.9 + 92.2i)T^{2} \)
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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

−10.58634817820898950920047536407, −9.418909488889359707284725781807, −8.324051648677829343580090544373, −6.91374457988725700508835678017, −6.68555960083195773325208141164, −5.39127578975700232303482145040, −4.61702133173576985853916977919, −3.45325297846759451621078912370, −2.66284536325453045022685457567, −1.70989293089169840382479691044, 2.51039781284536467884713288315, 3.33506835249043110106038932642, 4.06938399010728188957953056244, 4.98140885627862960847291812460, 6.12574942693229383220486091593, 6.54509665485608967977445155553, 7.83372305623039664852178828868, 8.431112386841033600886507085480, 9.806083882672218910534023725975, 10.57331748149476826666970656652

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