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Properties

Label 2-775-31.16-c1-0-37
Degree $2$
Conductor $775$
Sign $0.442 + 0.897i$
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  + (1.56 − 1.13i)2-s + (0.836 + 0.607i)3-s + (0.532 − 1.63i)4-s + 1.99·6-s + (0.394 − 1.21i)7-s + (0.165 + 0.508i)8-s + (−0.596 − 1.83i)9-s + (1.64 − 5.06i)11-s + (1.44 − 1.04i)12-s + (0.223 + 0.162i)13-s + (−0.761 − 2.34i)14-s + (3.62 + 2.63i)16-s + (1.10 + 3.41i)17-s + (−3.01 − 2.18i)18-s + (1.65 − 1.20i)19-s + ⋯
L(s)  = 1  + (1.10 − 0.801i)2-s + (0.483 + 0.351i)3-s + (0.266 − 0.819i)4-s + 0.814·6-s + (0.149 − 0.458i)7-s + (0.0583 + 0.179i)8-s + (−0.198 − 0.611i)9-s + (0.496 − 1.52i)11-s + (0.416 − 0.302i)12-s + (0.0620 + 0.0450i)13-s + (−0.203 − 0.625i)14-s + (0.905 + 0.657i)16-s + (0.268 + 0.827i)17-s + (−0.710 − 0.515i)18-s + (0.379 − 0.275i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.442 + 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.442 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $0.442 + 0.897i$
Analytic conductor: \(6.18840\)
Root analytic conductor: \(2.48765\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{775} (326, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 775,\ (\ :1/2),\ 0.442 + 0.897i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.78820 - 1.73442i\)
\(L(\frac12)\) \(\approx\) \(2.78820 - 1.73442i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.88 + 2.66i)T \)
good2 \( 1 + (-1.56 + 1.13i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (-0.836 - 0.607i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + (-0.394 + 1.21i)T + (-5.66 - 4.11i)T^{2} \)
11 \( 1 + (-1.64 + 5.06i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-0.223 - 0.162i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.10 - 3.41i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.65 + 1.20i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-1.29 - 3.97i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (2.95 - 2.14i)T + (8.96 - 27.5i)T^{2} \)
37 \( 1 - 8.08T + 37T^{2} \)
41 \( 1 + (3.08 - 2.24i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + (7.93 - 5.76i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (-0.680 - 0.494i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.25 + 3.86i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-5.12 - 3.72i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + 10.5T + 61T^{2} \)
67 \( 1 + 14.4T + 67T^{2} \)
71 \( 1 + (2.37 + 7.30i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (2.58 - 7.96i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (0.897 + 2.76i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (8.59 - 6.24i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (1.06 - 3.28i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (4.07 - 12.5i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.41414181278678687874695716505, −9.372598257329810666103470747638, −8.588756593605267749087771771977, −7.69357757936689255732061226846, −6.25577037368891056404301939985, −5.56722045927687103911933726012, −4.31127889041486924143641175430, −3.55439790745151930957072803068, −3.00248857960786274179171391742, −1.33914256129394527012108561445, 1.85179115146004039617127178427, 3.07678101386193407697168873847, 4.36541398937086437946544913460, 5.07352845785415593063251813725, 5.94329340248620765077895210410, 7.16875042470926283752343591031, 7.38608839680422684555750283235, 8.556945275741630260342200000998, 9.513027513287353041800123190014, 10.37233280136339428991631159860

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