Properties

Label 2-76-19.13-c6-0-9
Degree $2$
Conductor $76$
Sign $-0.916 - 0.399i$
Analytic cond. $17.4841$
Root an. cond. $4.18140$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (17.6 − 48.4i)3-s + (11.8 + 67.3i)5-s + (−186. + 323. i)7-s + (−1.47e3 − 1.24e3i)9-s + (−462. − 801. i)11-s + (−1.08e3 − 2.97e3i)13-s + (3.47e3 + 612. i)15-s + (−6.88e3 + 5.77e3i)17-s + (−6.57e3 + 1.94e3i)19-s + (1.23e4 + 1.47e4i)21-s + (−209. + 1.19e3i)23-s + (1.02e4 − 3.74e3i)25-s + (−5.36e4 + 3.09e4i)27-s + (2.07e4 − 2.47e4i)29-s + (1.03e4 + 5.95e3i)31-s + ⋯
L(s)  = 1  + (0.653 − 1.79i)3-s + (0.0950 + 0.539i)5-s + (−0.545 + 0.944i)7-s + (−2.02 − 1.70i)9-s + (−0.347 − 0.601i)11-s + (−0.492 − 1.35i)13-s + (1.02 + 0.181i)15-s + (−1.40 + 1.17i)17-s + (−0.958 + 0.283i)19-s + (1.33 + 1.59i)21-s + (−0.0172 + 0.0978i)23-s + (0.658 − 0.239i)25-s + (−2.72 + 1.57i)27-s + (0.851 − 1.01i)29-s + (0.345 + 0.199i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.916 - 0.399i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.916 - 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.916 - 0.399i$
Analytic conductor: \(17.4841\)
Root analytic conductor: \(4.18140\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :3),\ -0.916 - 0.399i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.162066 + 0.776874i\)
\(L(\frac12)\) \(\approx\) \(0.162066 + 0.776874i\)
\(L(4)\) 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 \( 1 + (6.57e3 - 1.94e3i)T \)
good3 \( 1 + (-17.6 + 48.4i)T + (-558. - 468. i)T^{2} \)
5 \( 1 + (-11.8 - 67.3i)T + (-1.46e4 + 5.34e3i)T^{2} \)
7 \( 1 + (186. - 323. i)T + (-5.88e4 - 1.01e5i)T^{2} \)
11 \( 1 + (462. + 801. i)T + (-8.85e5 + 1.53e6i)T^{2} \)
13 \( 1 + (1.08e3 + 2.97e3i)T + (-3.69e6 + 3.10e6i)T^{2} \)
17 \( 1 + (6.88e3 - 5.77e3i)T + (4.19e6 - 2.37e7i)T^{2} \)
23 \( 1 + (209. - 1.19e3i)T + (-1.39e8 - 5.06e7i)T^{2} \)
29 \( 1 + (-2.07e4 + 2.47e4i)T + (-1.03e8 - 5.85e8i)T^{2} \)
31 \( 1 + (-1.03e4 - 5.95e3i)T + (4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 - 3.57e4iT - 2.56e9T^{2} \)
41 \( 1 + (-1.87e4 + 5.15e4i)T + (-3.63e9 - 3.05e9i)T^{2} \)
43 \( 1 + (6.29e3 + 3.56e4i)T + (-5.94e9 + 2.16e9i)T^{2} \)
47 \( 1 + (1.05e5 + 8.82e4i)T + (1.87e9 + 1.06e10i)T^{2} \)
53 \( 1 + (-1.15e5 - 2.03e4i)T + (2.08e10 + 7.58e9i)T^{2} \)
59 \( 1 + (1.26e4 + 1.50e4i)T + (-7.32e9 + 4.15e10i)T^{2} \)
61 \( 1 + (-3.76e4 + 2.13e5i)T + (-4.84e10 - 1.76e10i)T^{2} \)
67 \( 1 + (-1.79e5 + 2.14e5i)T + (-1.57e10 - 8.90e10i)T^{2} \)
71 \( 1 + (5.98e5 - 1.05e5i)T + (1.20e11 - 4.38e10i)T^{2} \)
73 \( 1 + (4.44e5 + 1.61e5i)T + (1.15e11 + 9.72e10i)T^{2} \)
79 \( 1 + (-1.74e5 + 4.78e5i)T + (-1.86e11 - 1.56e11i)T^{2} \)
83 \( 1 + (-1.54e5 + 2.67e5i)T + (-1.63e11 - 2.83e11i)T^{2} \)
89 \( 1 + (-1.56e5 - 4.30e5i)T + (-3.80e11 + 3.19e11i)T^{2} \)
97 \( 1 + (2.42e5 + 2.88e5i)T + (-1.44e11 + 8.20e11i)T^{2} \)
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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

−12.85387409145510382737457451740, −12.00122299125096472083379327176, −10.52675665679794342948323683354, −8.726509972454616833809971537028, −8.097294409999198488792961705765, −6.65966799085407755191092139272, −5.95731285021099647445770424491, −3.01914705096005886035402725391, −2.17885065866138184344364491708, −0.25200198086438055077545479430, 2.63048454100789509984924965079, 4.30420752355223280428819960240, 4.75444189122888540660027127243, 6.95083745187990649050255209408, 8.724263773238954532397093575676, 9.426901596548978301373750868858, 10.33337253500383377900913780259, 11.34486947654432330175224338065, 13.11070430701847785235212258937, 14.09615452873344602698102706878

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