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Properties

Label 2-76-19.8-c2-0-0
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: $\chi_{76} (65, \cdot )$
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

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

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