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Properties

Label 2-76-19.12-c4-0-4
Degree $2$
Conductor $76$
Sign $0.587 + 0.809i$
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  + (12.4 − 7.18i)3-s + (−3.11 − 5.39i)5-s + 49.9·7-s + (62.6 − 108. i)9-s + 1.88·11-s + (−69.0 − 39.8i)13-s + (−77.4 − 44.7i)15-s + (−119. − 207. i)17-s + (72.4 + 353. i)19-s + (621. − 358. i)21-s + (109. − 189. i)23-s + (293. − 507. i)25-s − 635. i·27-s + (340. + 196. i)29-s + 580. i·31-s + ⋯
L(s)  = 1  + (1.38 − 0.797i)3-s + (−0.124 − 0.215i)5-s + 1.01·7-s + (0.773 − 1.33i)9-s + 0.0155·11-s + (−0.408 − 0.235i)13-s + (−0.344 − 0.198i)15-s + (−0.414 − 0.717i)17-s + (0.200 + 0.979i)19-s + (1.40 − 0.813i)21-s + (0.206 − 0.358i)23-s + (0.468 − 0.812i)25-s − 0.872i·27-s + (0.404 + 0.233i)29-s + 0.604i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.587 + 0.809i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.587 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.587 + 0.809i$
Analytic conductor: \(7.85611\)
Root analytic conductor: \(2.80287\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :2),\ 0.587 + 0.809i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.34603 - 1.19608i\)
\(L(\frac12)\) \(\approx\) \(2.34603 - 1.19608i\)
\(L(3)\) 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 \( 1 + (-72.4 - 353. i)T \)
good3 \( 1 + (-12.4 + 7.18i)T + (40.5 - 70.1i)T^{2} \)
5 \( 1 + (3.11 + 5.39i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 - 49.9T + 2.40e3T^{2} \)
11 \( 1 - 1.88T + 1.46e4T^{2} \)
13 \( 1 + (69.0 + 39.8i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (119. + 207. i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (-109. + 189. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-340. - 196. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 - 580. iT - 9.23e5T^{2} \)
37 \( 1 - 2.47e3iT - 1.87e6T^{2} \)
41 \( 1 + (2.84e3 - 1.64e3i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (162. + 282. i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (969. - 1.67e3i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (-2.08e3 - 1.20e3i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (-3.04e3 + 1.75e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (1.03e3 - 1.79e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (4.38e3 + 2.53e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (-2.99e3 + 1.72e3i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (-4.35e3 - 7.54e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (-3.64e3 + 2.10e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + 1.17e4T + 4.74e7T^{2} \)
89 \( 1 + (8.21e3 + 4.74e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (-5.04e3 + 2.91e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−13.80360658248663230030146658196, −12.67724737845116263110657817980, −11.66643784593407349381282140834, −10.03686755491720233690935996718, −8.601590377832472124611686536989, −8.047887610493371955320626868860, −6.85401154415851992627133978375, −4.77370842077721846772393869341, −2.94038208752655051786735762544, −1.46812415398660702429757093220, 2.19017087868449954910194411263, 3.73178270244717818420433490676, 4.98576808351448359630920904634, 7.24350237941195324026177638938, 8.397949433010437402242127503738, 9.225585404360674939440439715450, 10.46204227292791587309959059521, 11.53023996657602433446095914694, 13.20114445356397371752443567306, 14.18658386404268929416490063797

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