Properties

Label 2-76-19.15-c4-0-6
Degree $2$
Conductor $76$
Sign $-0.870 + 0.492i$
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  + (0.693 − 0.122i)3-s + (−24.2 + 20.3i)5-s + (32.5 − 56.4i)7-s + (−75.6 + 27.5i)9-s + (−96.0 − 166. i)11-s + (−85.9 − 15.1i)13-s + (−14.3 + 17.0i)15-s + (−103. − 37.5i)17-s + (−84.7 + 350. i)19-s + (15.6 − 43.1i)21-s + (−676. − 567. i)23-s + (64.9 − 368. i)25-s + (−98.4 + 56.8i)27-s + (−495. − 1.36e3i)29-s + (1.38e3 + 801. i)31-s + ⋯
L(s)  = 1  + (0.0770 − 0.0135i)3-s + (−0.968 + 0.812i)5-s + (0.664 − 1.15i)7-s + (−0.933 + 0.339i)9-s + (−0.794 − 1.37i)11-s + (−0.508 − 0.0896i)13-s + (−0.0635 + 0.0757i)15-s + (−0.356 − 0.129i)17-s + (−0.234 + 0.972i)19-s + (0.0355 − 0.0977i)21-s + (−1.27 − 1.07i)23-s + (0.103 − 0.589i)25-s + (−0.135 + 0.0779i)27-s + (−0.589 − 1.61i)29-s + (1.44 + 0.833i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.870 + 0.492i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.870 + 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.870 + 0.492i$
Analytic conductor: \(7.85611\)
Root analytic conductor: \(2.80287\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :2),\ -0.870 + 0.492i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0928342 - 0.352265i\)
\(L(\frac12)\) \(\approx\) \(0.0928342 - 0.352265i\)
\(L(3)\) 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 \( 1 + (84.7 - 350. i)T \)
good3 \( 1 + (-0.693 + 0.122i)T + (76.1 - 27.7i)T^{2} \)
5 \( 1 + (24.2 - 20.3i)T + (108. - 615. i)T^{2} \)
7 \( 1 + (-32.5 + 56.4i)T + (-1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (96.0 + 166. i)T + (-7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (85.9 + 15.1i)T + (2.68e4 + 9.76e3i)T^{2} \)
17 \( 1 + (103. + 37.5i)T + (6.39e4 + 5.36e4i)T^{2} \)
23 \( 1 + (676. + 567. i)T + (4.85e4 + 2.75e5i)T^{2} \)
29 \( 1 + (495. + 1.36e3i)T + (-5.41e5 + 4.54e5i)T^{2} \)
31 \( 1 + (-1.38e3 - 801. i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 - 2.39e3iT - 1.87e6T^{2} \)
41 \( 1 + (-2.31e3 + 407. i)T + (2.65e6 - 9.66e5i)T^{2} \)
43 \( 1 + (-175. + 147. i)T + (5.93e5 - 3.36e6i)T^{2} \)
47 \( 1 + (-344. + 125. i)T + (3.73e6 - 3.13e6i)T^{2} \)
53 \( 1 + (2.07e3 - 2.47e3i)T + (-1.37e6 - 7.77e6i)T^{2} \)
59 \( 1 + (624. - 1.71e3i)T + (-9.28e6 - 7.78e6i)T^{2} \)
61 \( 1 + (2.11e3 + 1.77e3i)T + (2.40e6 + 1.36e7i)T^{2} \)
67 \( 1 + (2.01e3 + 5.52e3i)T + (-1.54e7 + 1.29e7i)T^{2} \)
71 \( 1 + (-876. - 1.04e3i)T + (-4.41e6 + 2.50e7i)T^{2} \)
73 \( 1 + (985. + 5.58e3i)T + (-2.66e7 + 9.71e6i)T^{2} \)
79 \( 1 + (4.62e3 - 814. i)T + (3.66e7 - 1.33e7i)T^{2} \)
83 \( 1 + (375. - 650. i)T + (-2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 + (7.66e3 + 1.35e3i)T + (5.89e7 + 2.14e7i)T^{2} \)
97 \( 1 + (-2.18e3 + 6.01e3i)T + (-6.78e7 - 5.69e7i)T^{2} \)
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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

−13.72318926381830726449353983485, −11.92994205174136844946931738085, −11.01984886864649136329669694006, −10.36651888157801358718300998752, −8.166808013230803464710184087820, −7.81138521383791335446348994646, −6.14290777274139884386216855174, −4.33834632717160852025310469361, −2.90406344736900152765420630377, −0.17436763622985201307717525829, 2.34658841497915221674036812290, 4.44819977752285628253883140832, 5.53227441982605638769665028669, 7.52642002088543364733212518222, 8.490631051591991882863731449501, 9.456344557753523439111444537953, 11.25290533496110802803477535243, 12.07243153734660955941166100414, 12.79721694701081704233716768641, 14.48387859626889835531851101749

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