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Properties

Label 2-76-4.3-c2-0-13
Degree $2$
Conductor $76$
Sign $-0.0607 + 0.998i$
Analytic cond. $2.07085$
Root an. cond. $1.43904$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0607 + 1.99i)2-s − 5.37i·3-s + (−3.99 − 0.242i)4-s − 5.82·5-s + (10.7 + 0.326i)6-s − 5.45i·7-s + (0.728 − 7.96i)8-s − 19.9·9-s + (0.353 − 11.6i)10-s − 1.60i·11-s + (−1.30 + 21.4i)12-s + 23.5·13-s + (10.8 + 0.331i)14-s + 31.3i·15-s + (15.8 + 1.94i)16-s − 5.92·17-s + ⋯
L(s)  = 1  + (−0.0303 + 0.999i)2-s − 1.79i·3-s + (−0.998 − 0.0607i)4-s − 1.16·5-s + (1.79 + 0.0544i)6-s − 0.778i·7-s + (0.0910 − 0.995i)8-s − 2.21·9-s + (0.0353 − 1.16i)10-s − 0.146i·11-s + (−0.108 + 1.78i)12-s + 1.80·13-s + (0.778 + 0.0236i)14-s + 2.08i·15-s + (0.992 + 0.121i)16-s − 0.348·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0607 + 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0607 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.0607 + 0.998i$
Analytic conductor: \(2.07085\)
Root analytic conductor: \(1.43904\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1),\ -0.0607 + 0.998i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.534833 - 0.568370i\)
\(L(\frac12)\) \(\approx\) \(0.534833 - 0.568370i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0607 - 1.99i)T \)
19 \( 1 - 4.35iT \)
good3 \( 1 + 5.37iT - 9T^{2} \)
5 \( 1 + 5.82T + 25T^{2} \)
7 \( 1 + 5.45iT - 49T^{2} \)
11 \( 1 + 1.60iT - 121T^{2} \)
13 \( 1 - 23.5T + 169T^{2} \)
17 \( 1 + 5.92T + 289T^{2} \)
23 \( 1 + 26.6iT - 529T^{2} \)
29 \( 1 + 1.49T + 841T^{2} \)
31 \( 1 + 31.3iT - 961T^{2} \)
37 \( 1 - 26.8T + 1.36e3T^{2} \)
41 \( 1 + 44.0T + 1.68e3T^{2} \)
43 \( 1 + 27.8iT - 1.84e3T^{2} \)
47 \( 1 - 32.5iT - 2.20e3T^{2} \)
53 \( 1 - 76.7T + 2.80e3T^{2} \)
59 \( 1 - 33.8iT - 3.48e3T^{2} \)
61 \( 1 - 53.0T + 3.72e3T^{2} \)
67 \( 1 + 76.1iT - 4.48e3T^{2} \)
71 \( 1 + 59.9iT - 5.04e3T^{2} \)
73 \( 1 + 49.8T + 5.32e3T^{2} \)
79 \( 1 - 23.3iT - 6.24e3T^{2} \)
83 \( 1 - 137. iT - 6.88e3T^{2} \)
89 \( 1 - 116.T + 7.92e3T^{2} \)
97 \( 1 + 65.7T + 9.40e3T^{2} \)
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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

−13.73286683528631366153765071728, −13.25400723609121671165070542546, −12.09841502768406806587730542996, −10.92458425363601150563000757387, −8.602233552729455584144185441316, −7.960146630577004036951313767473, −7.00524092322590044734123891049, −6.08373680702418033094449666325, −3.90352909334339499873127467989, −0.72099987330905502382450209364, 3.32541785713982167546021892250, 4.15887494973096110744644782275, 5.48179916046875205790189474457, 8.402034392153080377471842206311, 9.026314259659546360153594681789, 10.27339782917484744590981006188, 11.29210774393287323246159421883, 11.74695678075890719116852727257, 13.39885866975245257235512754916, 14.79987553268109375800197592405

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