Properties

Label 2-76-76.7-c2-0-16
Degree $2$
Conductor $76$
Sign $-0.868 + 0.496i$
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  + (1.12 − 1.65i)2-s + (−3.58 + 2.06i)3-s + (−1.46 − 3.72i)4-s + (−3.72 − 6.46i)5-s + (−0.611 + 8.24i)6-s − 3.06i·7-s + (−7.80 − 1.76i)8-s + (4.05 − 7.02i)9-s + (−14.8 − 1.10i)10-s + 6.31i·11-s + (12.9 + 10.2i)12-s + (8.74 − 15.1i)13-s + (−5.07 − 3.45i)14-s + (26.7 + 15.4i)15-s + (−11.6 + 10.9i)16-s + (10.6 + 18.4i)17-s + ⋯
L(s)  = 1  + (0.562 − 0.826i)2-s + (−1.19 + 0.689i)3-s + (−0.366 − 0.930i)4-s + (−0.745 − 1.29i)5-s + (−0.101 + 1.37i)6-s − 0.438i·7-s + (−0.975 − 0.220i)8-s + (0.450 − 0.780i)9-s + (−1.48 − 0.110i)10-s + 0.574i·11-s + (1.07 + 0.857i)12-s + (0.672 − 1.16i)13-s + (−0.362 − 0.246i)14-s + (1.78 + 1.02i)15-s + (−0.730 + 0.682i)16-s + (0.628 + 1.08i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.868 + 0.496i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.868 + 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.868 + 0.496i$
Analytic conductor: \(2.07085\)
Root analytic conductor: \(1.43904\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1),\ -0.868 + 0.496i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.199746 - 0.752326i\)
\(L(\frac12)\) \(\approx\) \(0.199746 - 0.752326i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.12 + 1.65i)T \)
19 \( 1 + (6.62 + 17.8i)T \)
good3 \( 1 + (3.58 - 2.06i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (3.72 + 6.46i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + 3.06iT - 49T^{2} \)
11 \( 1 - 6.31iT - 121T^{2} \)
13 \( 1 + (-8.74 + 15.1i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (-10.6 - 18.4i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (-5.19 - 3.00i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-18.2 + 31.6i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + 53.3iT - 961T^{2} \)
37 \( 1 + 39.2T + 1.36e3T^{2} \)
41 \( 1 + (-18.2 - 31.6i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (31.7 - 18.3i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (0.0577 + 0.0333i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (31.6 - 54.8i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-41.8 + 24.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-29.8 + 51.7i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-58.4 - 33.7i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (30.0 - 17.3i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (17.7 + 30.7i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-65.0 + 37.5i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 53.5iT - 6.88e3T^{2} \)
89 \( 1 + (-36.5 + 63.2i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (59.7 + 103. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−13.29970986889944861360611291354, −12.56038907561608244395633476645, −11.62723783668560554721182954602, −10.73467093519198243451893154004, −9.768217067107461739097318174639, −8.234254151732862831554303523811, −5.95513855885430612525409261624, −4.85883489707286642098394632497, −3.99008265770114378100079956491, −0.66454454164952640190629966802, 3.44132931462387618244787056588, 5.36339983024465513428245679640, 6.60482388599402902597950903750, 7.10148312094764087246016712454, 8.617809982951301149808386093076, 10.77708887006945014590806676580, 11.76547010680136646375124472847, 12.29073029062182342673757822837, 13.87678021975612928659827213387, 14.58665684245283522262074822164

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