Properties

Label 2-76-76.11-c2-0-3
Degree $2$
Conductor $76$
Sign $-0.643 - 0.765i$
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  + (−0.784 + 1.83i)2-s + (1.67 + 0.967i)3-s + (−2.76 − 2.88i)4-s + (−1.80 + 3.12i)5-s + (−3.09 + 2.32i)6-s + 13.2i·7-s + (7.48 − 2.82i)8-s + (−2.62 − 4.55i)9-s + (−4.33 − 5.77i)10-s + 2.62i·11-s + (−1.84 − 7.51i)12-s + (−0.424 − 0.735i)13-s + (−24.3 − 10.3i)14-s + (−6.04 + 3.49i)15-s + (−0.676 + 15.9i)16-s + (6.24 − 10.8i)17-s + ⋯
L(s)  = 1  + (−0.392 + 0.919i)2-s + (0.558 + 0.322i)3-s + (−0.691 − 0.721i)4-s + (−0.361 + 0.625i)5-s + (−0.515 + 0.387i)6-s + 1.89i·7-s + (0.935 − 0.353i)8-s + (−0.292 − 0.505i)9-s + (−0.433 − 0.577i)10-s + 0.239i·11-s + (−0.153 − 0.626i)12-s + (−0.0326 − 0.0565i)13-s + (−1.73 − 0.742i)14-s + (−0.403 + 0.232i)15-s + (−0.0423 + 0.999i)16-s + (0.367 − 0.636i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.643 - 0.765i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.643 - 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.643 - 0.765i$
Analytic conductor: \(2.07085\)
Root analytic conductor: \(1.43904\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1),\ -0.643 - 0.765i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.441529 + 0.948096i\)
\(L(\frac12)\) \(\approx\) \(0.441529 + 0.948096i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.784 - 1.83i)T \)
19 \( 1 + (-18.2 - 5.21i)T \)
good3 \( 1 + (-1.67 - 0.967i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (1.80 - 3.12i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 - 13.2iT - 49T^{2} \)
11 \( 1 - 2.62iT - 121T^{2} \)
13 \( 1 + (0.424 + 0.735i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (-6.24 + 10.8i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + (-26.9 + 15.5i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (9.98 + 17.2i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 - 28.0iT - 961T^{2} \)
37 \( 1 - 61.9T + 1.36e3T^{2} \)
41 \( 1 + (-10.4 + 18.1i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (23.9 + 13.8i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (44.2 - 25.5i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-9.57 - 16.5i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (63.8 + 36.8i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (32.5 + 56.4i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (21.6 - 12.4i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (-32.3 - 18.6i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-15.9 + 27.5i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-82.1 - 47.4i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 22.6iT - 6.88e3T^{2} \)
89 \( 1 + (-60.9 - 105. i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (10.1 - 17.6i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−15.02626215671879811753064301453, −14.14553091176347710587589279445, −12.51767363218780717464335846272, −11.31617620513177970375027752533, −9.611782737198953361822968468645, −9.000991403733892791448569269437, −7.85989399090247068126071216852, −6.41599797767734116167185569121, −5.16854207138260445078897788154, −3.00864095533965611163129957826, 1.09079327761329753521378846179, 3.35300886027746358872899340491, 4.68765605354351079912945002416, 7.41537913463213902815304123545, 8.090508710187013147081819207754, 9.409372848231635992278274050493, 10.62728287651599797788614967279, 11.48522230879408211781780698668, 13.05565281379222189838070340750, 13.44820102972602938846013972646

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