Properties

Label 2-76-19.3-c4-0-3
Degree $2$
Conductor $76$
Sign $0.951 + 0.307i$
Analytic cond. $7.85611$
Root an. cond. $2.80287$
Motivic weight $4$
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 − 6.28i)3-s + (−2.03 + 11.5i)5-s + (38.6 + 66.9i)7-s + (27.7 − 23.3i)9-s + (79.3 − 137. i)11-s + (65.1 − 179. i)13-s + (77.0 − 13.5i)15-s + (343. + 288. i)17-s + (−191. − 306. i)19-s + (332. − 395. i)21-s + (99.9 + 567. i)23-s + (458. + 167. i)25-s + (−679. − 392. i)27-s + (447. + 533. i)29-s + (533. − 308. i)31-s + ⋯
L(s)  = 1  + (−0.254 − 0.698i)3-s + (−0.0812 + 0.460i)5-s + (0.788 + 1.36i)7-s + (0.343 − 0.287i)9-s + (0.655 − 1.13i)11-s + (0.385 − 1.05i)13-s + (0.342 − 0.0603i)15-s + (1.18 + 0.997i)17-s + (−0.530 − 0.847i)19-s + (0.753 − 0.897i)21-s + (0.189 + 1.07i)23-s + (0.734 + 0.267i)25-s + (−0.931 − 0.537i)27-s + (0.531 + 0.633i)29-s + (0.555 − 0.320i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.307i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.951 + 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.951 + 0.307i$
Analytic conductor: \(7.85611\)
Root analytic conductor: \(2.80287\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :2),\ 0.951 + 0.307i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.72656 - 0.272352i\)
\(L(\frac12)\) \(\approx\) \(1.72656 - 0.272352i\)
\(L(3)\) 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 \( 1 + (191. + 306. i)T \)
good3 \( 1 + (2.28 + 6.28i)T + (-62.0 + 52.0i)T^{2} \)
5 \( 1 + (2.03 - 11.5i)T + (-587. - 213. i)T^{2} \)
7 \( 1 + (-38.6 - 66.9i)T + (-1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (-79.3 + 137. i)T + (-7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (-65.1 + 179. i)T + (-2.18e4 - 1.83e4i)T^{2} \)
17 \( 1 + (-343. - 288. i)T + (1.45e4 + 8.22e4i)T^{2} \)
23 \( 1 + (-99.9 - 567. i)T + (-2.62e5 + 9.57e4i)T^{2} \)
29 \( 1 + (-447. - 533. i)T + (-1.22e5 + 6.96e5i)T^{2} \)
31 \( 1 + (-533. + 308. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 - 2.28e3iT - 1.87e6T^{2} \)
41 \( 1 + (792. + 2.17e3i)T + (-2.16e6 + 1.81e6i)T^{2} \)
43 \( 1 + (-465. + 2.64e3i)T + (-3.21e6 - 1.16e6i)T^{2} \)
47 \( 1 + (1.39e3 - 1.17e3i)T + (8.47e5 - 4.80e6i)T^{2} \)
53 \( 1 + (1.88e3 - 333. i)T + (7.41e6 - 2.69e6i)T^{2} \)
59 \( 1 + (1.32e3 - 1.57e3i)T + (-2.10e6 - 1.19e7i)T^{2} \)
61 \( 1 + (-228. - 1.29e3i)T + (-1.30e7 + 4.73e6i)T^{2} \)
67 \( 1 + (3.21e3 + 3.83e3i)T + (-3.49e6 + 1.98e7i)T^{2} \)
71 \( 1 + (4.66e3 + 822. i)T + (2.38e7 + 8.69e6i)T^{2} \)
73 \( 1 + (-6.55e3 + 2.38e3i)T + (2.17e7 - 1.82e7i)T^{2} \)
79 \( 1 + (2.01e3 + 5.54e3i)T + (-2.98e7 + 2.50e7i)T^{2} \)
83 \( 1 + (-2.41e3 - 4.18e3i)T + (-2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 + (1.67e3 - 4.59e3i)T + (-4.80e7 - 4.03e7i)T^{2} \)
97 \( 1 + (-42.3 + 50.4i)T + (-1.53e7 - 8.71e7i)T^{2} \)
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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

−13.63764586348847021901129766487, −12.45269088375045711422485903202, −11.69207363949535112484114724275, −10.61484290731938919335544776312, −8.916183589874714459625888955376, −7.964282004606106910010278131303, −6.44492105796614185342866769837, −5.47046453987085750140404943025, −3.22499536306591290953231582320, −1.29532781544470612463841250095, 1.34868005748958072092044441039, 4.21260213949640509162219283871, 4.72643604028979589550983472784, 6.82637350980586579714073286093, 7.989558113072451761923058130935, 9.569213178620163461409398253910, 10.40624341883946621441936732158, 11.49989613424743086644683866535, 12.65907962776248098638691979536, 14.07904025366318278104830376531

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