Properties

Label 2-76-19.8-c6-0-7
Degree $2$
Conductor $76$
Sign $-0.615 + 0.787i$
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  + (−31.9 − 18.4i)3-s + (71.3 − 123. i)5-s + 421.·7-s + (315. + 545. i)9-s + 1.06e3·11-s + (99.2 − 57.3i)13-s + (−4.55e3 + 2.63e3i)15-s + (1.34e3 − 2.33e3i)17-s + (1.61e3 − 6.66e3i)19-s + (−1.34e4 − 7.77e3i)21-s + (−8.93e3 − 1.54e4i)23-s + (−2.36e3 − 4.09e3i)25-s + 3.64e3i·27-s + (−2.53e4 + 1.46e4i)29-s − 4.06e4i·31-s + ⋯
L(s)  = 1  + (−1.18 − 0.682i)3-s + (0.570 − 0.988i)5-s + 1.22·7-s + (0.432 + 0.748i)9-s + 0.800·11-s + (0.0451 − 0.0260i)13-s + (−1.34 + 0.779i)15-s + (0.274 − 0.474i)17-s + (0.235 − 0.971i)19-s + (−1.45 − 0.839i)21-s + (−0.734 − 1.27i)23-s + (−0.151 − 0.262i)25-s + 0.184i·27-s + (−1.03 + 0.599i)29-s − 1.36i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 + 0.787i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.615 + 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.615 + 0.787i$
Analytic conductor: \(17.4841\)
Root analytic conductor: \(4.18140\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :3),\ -0.615 + 0.787i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.620286 - 1.27200i\)
\(L(\frac12)\) \(\approx\) \(0.620286 - 1.27200i\)
\(L(4)\) 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 \( 1 + (-1.61e3 + 6.66e3i)T \)
good3 \( 1 + (31.9 + 18.4i)T + (364.5 + 631. i)T^{2} \)
5 \( 1 + (-71.3 + 123. i)T + (-7.81e3 - 1.35e4i)T^{2} \)
7 \( 1 - 421.T + 1.17e5T^{2} \)
11 \( 1 - 1.06e3T + 1.77e6T^{2} \)
13 \( 1 + (-99.2 + 57.3i)T + (2.41e6 - 4.18e6i)T^{2} \)
17 \( 1 + (-1.34e3 + 2.33e3i)T + (-1.20e7 - 2.09e7i)T^{2} \)
23 \( 1 + (8.93e3 + 1.54e4i)T + (-7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 + (2.53e4 - 1.46e4i)T + (2.97e8 - 5.15e8i)T^{2} \)
31 \( 1 + 4.06e4iT - 8.87e8T^{2} \)
37 \( 1 - 7.35e4iT - 2.56e9T^{2} \)
41 \( 1 + (-2.25e4 - 1.30e4i)T + (2.37e9 + 4.11e9i)T^{2} \)
43 \( 1 + (1.77e3 - 3.07e3i)T + (-3.16e9 - 5.47e9i)T^{2} \)
47 \( 1 + (7.49e4 + 1.29e5i)T + (-5.38e9 + 9.33e9i)T^{2} \)
53 \( 1 + (-7.94e3 + 4.58e3i)T + (1.10e10 - 1.91e10i)T^{2} \)
59 \( 1 + (2.11e5 + 1.22e5i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (-5.70e4 - 9.88e4i)T + (-2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (2.60e4 - 1.50e4i)T + (4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 + (-4.70e5 - 2.71e5i)T + (6.40e10 + 1.10e11i)T^{2} \)
73 \( 1 + (-1.96e5 + 3.40e5i)T + (-7.56e10 - 1.31e11i)T^{2} \)
79 \( 1 + (-5.59e5 - 3.22e5i)T + (1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + 4.65e5T + 3.26e11T^{2} \)
89 \( 1 + (5.01e5 - 2.89e5i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 + (3.10e5 + 1.79e5i)T + (4.16e11 + 7.21e11i)T^{2} \)
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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

−12.73101186561640531012095688553, −11.78683325143315904747455790207, −11.10236703202217403038721233238, −9.472271742386873908428024475209, −8.230657605977869328324140708224, −6.78805957214541319966038983757, −5.54819183403847096617288053957, −4.65855252207277467384994108450, −1.71372195301612806596437219452, −0.67038410425217523208772420823, 1.65716064563745683917984983411, 3.89547094566980246782767159709, 5.36373394891363148722299532016, 6.25091258563248474006348492813, 7.77070522490497652717578971924, 9.549218462663661096764872305086, 10.61080158336478249980478241172, 11.25230877977761386150285604883, 12.18164682828769330116200355028, 14.05688623497516190128769941634

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