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Properties

Label 2-76-19.12-c2-0-3
Degree $2$
Conductor $76$
Sign $0.516 + 0.856i$
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  + (2.28 − 1.32i)3-s + (−4.81 − 8.34i)5-s + 7.59·7-s + (−1.00 + 1.74i)9-s − 4.61·11-s + (19.8 + 11.4i)13-s + (−22.0 − 12.7i)15-s + (3.02 + 5.23i)17-s + (−2.21 − 18.8i)19-s + (17.3 − 10.0i)21-s + (−11.8 + 20.5i)23-s + (−33.9 + 58.7i)25-s + 29.1i·27-s + (10.1 + 5.87i)29-s − 4.22i·31-s + ⋯
L(s)  = 1  + (0.762 − 0.440i)3-s + (−0.963 − 1.66i)5-s + 1.08·7-s + (−0.111 + 0.193i)9-s − 0.419·11-s + (1.52 + 0.880i)13-s + (−1.46 − 0.848i)15-s + (0.177 + 0.307i)17-s + (−0.116 − 0.993i)19-s + (0.827 − 0.477i)21-s + (−0.515 + 0.893i)23-s + (−1.35 + 2.34i)25-s + 1.07i·27-s + (0.351 + 0.202i)29-s − 0.136i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.516 + 0.856i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.516 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.516 + 0.856i$
Analytic conductor: \(2.07085\)
Root analytic conductor: \(1.43904\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1),\ 0.516 + 0.856i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.26716 - 0.715453i\)
\(L(\frac12)\) \(\approx\) \(1.26716 - 0.715453i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (2.21 + 18.8i)T \)
good3 \( 1 + (-2.28 + 1.32i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (4.81 + 8.34i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 - 7.59T + 49T^{2} \)
11 \( 1 + 4.61T + 121T^{2} \)
13 \( 1 + (-19.8 - 11.4i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (-3.02 - 5.23i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (11.8 - 20.5i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-10.1 - 5.87i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + 4.22iT - 961T^{2} \)
37 \( 1 + 11.8iT - 1.36e3T^{2} \)
41 \( 1 + (7.64 - 4.41i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (11.5 + 19.9i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-33.0 + 57.3i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-40.7 - 23.5i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (7.52 - 4.34i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (3.57 - 6.19i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (77.3 + 44.6i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (67.9 - 39.2i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-43.1 - 74.6i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-57.9 + 33.4i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 5.30T + 6.88e3T^{2} \)
89 \( 1 + (99.6 + 57.5i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (49.2 - 28.4i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−13.79600720311198521918835772757, −13.20764032959465266666483011080, −11.92646879544716362951017954129, −11.10130549524989027264137822114, −8.901695671475969434297363655746, −8.453280311257751569818198915880, −7.55269862907470285038029409801, −5.27043102968662751569411223288, −4.03203682924524694802977894758, −1.52046694527998206135774905193, 2.93472957080253027791520474420, 4.00708381771493063989706295936, 6.20184277996454997315067131359, 7.82023761086843678010217472766, 8.376230331931387485809705964642, 10.27485621248281320993839097887, 10.96803030803421662996212430885, 11.99544778778419114792235386713, 13.84926696611659000383982656133, 14.59821686859137065472571983724

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