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Properties

Label 2-774-129.128-c1-0-4
Degree $2$
Conductor $774$
Sign $0.678 - 0.734i$
Analytic cond. $6.18042$
Root an. cond. $2.48604$
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-s + 4-s + 0.705·5-s + 3.39i·7-s + 8-s + 0.705·10-s + 1.97i·11-s + 1.29·13-s + 3.39i·14-s + 16-s − 1.41i·17-s + 4.38i·19-s + 0.705·20-s + 1.97i·22-s − 0.416i·23-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 0.315·5-s + 1.28i·7-s + 0.353·8-s + 0.223·10-s + 0.596i·11-s + 0.358·13-s + 0.906i·14-s + 0.250·16-s − 0.342i·17-s + 1.00i·19-s + 0.157·20-s + 0.421i·22-s − 0.0867i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 774 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 774 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(774\)    =    \(2 \cdot 3^{2} \cdot 43\)
Sign: $0.678 - 0.734i$
Analytic conductor: \(6.18042\)
Root analytic conductor: \(2.48604\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{774} (773, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 774,\ (\ :1/2),\ 0.678 - 0.734i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.27239 + 0.994609i\)
\(L(\frac12)\) \(\approx\) \(2.27239 + 0.994609i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
43 \( 1 + (-6.50 - 0.851i)T \)
good5 \( 1 - 0.705T + 5T^{2} \)
7 \( 1 - 3.39iT - 7T^{2} \)
11 \( 1 - 1.97iT - 11T^{2} \)
13 \( 1 - 1.29T + 13T^{2} \)
17 \( 1 + 1.41iT - 17T^{2} \)
19 \( 1 - 4.38iT - 19T^{2} \)
23 \( 1 + 0.416iT - 23T^{2} \)
29 \( 1 - 8.88T + 29T^{2} \)
31 \( 1 + 1.38T + 31T^{2} \)
37 \( 1 - 1.99iT - 37T^{2} \)
41 \( 1 + 7.19iT - 41T^{2} \)
47 \( 1 - 0.979iT - 47T^{2} \)
53 \( 1 + 9.19iT - 53T^{2} \)
59 \( 1 + 9.19iT - 59T^{2} \)
61 \( 1 - 7.48iT - 61T^{2} \)
67 \( 1 + 2.79T + 67T^{2} \)
71 \( 1 - 4.79T + 71T^{2} \)
73 \( 1 + 5.78iT - 73T^{2} \)
79 \( 1 + 9.00T + 79T^{2} \)
83 \( 1 + 0.0188iT - 83T^{2} \)
89 \( 1 - 5.00T + 89T^{2} \)
97 \( 1 + 10.9T + 97T^{2} \)
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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

−10.41649100649983029081959065307, −9.648083315571217373995232885213, −8.707795939405186579983566212089, −7.83904692553567150873766260728, −6.67830176501590520131530280013, −5.87074708673183921900552498680, −5.17814985098532774740371753619, −4.05374658423841465472095496409, −2.80721933771434458423389176802, −1.83667511618657926328826171015, 1.09475055647613309986857902867, 2.72022778789686671646109721655, 3.84874611012346321011630518176, 4.62434031570442658463673137988, 5.79675625122644534158747293615, 6.59603109006036674023499712917, 7.44630545244150022265482194543, 8.365903629056692690951084065175, 9.479050729283594375739141498742, 10.46813853017592541779626963058

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