Properties

Label 2-76-19.2-c4-0-3
Degree $2$
Conductor $76$
Sign $-0.168 + 0.985i$
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  + (−8.26 + 9.85i)3-s + (−19.2 + 6.99i)5-s + (1.56 + 2.70i)7-s + (−14.6 − 83.1i)9-s + (57.2 − 99.1i)11-s + (−73.1 − 87.2i)13-s + (89.9 − 247. i)15-s + (11.6 − 66.0i)17-s + (113. − 342. i)19-s + (−39.5 − 6.97i)21-s + (−229. − 83.6i)23-s + (−158. + 133. i)25-s + (37.8 + 21.8i)27-s + (−1.36e3 + 240. i)29-s + (708. − 408. i)31-s + ⋯
L(s)  = 1  + (−0.918 + 1.09i)3-s + (−0.768 + 0.279i)5-s + (0.0318 + 0.0551i)7-s + (−0.180 − 1.02i)9-s + (0.472 − 0.819i)11-s + (−0.433 − 0.516i)13-s + (0.399 − 1.09i)15-s + (0.0402 − 0.228i)17-s + (0.314 − 0.949i)19-s + (−0.0896 − 0.0158i)21-s + (−0.434 − 0.158i)23-s + (−0.253 + 0.212i)25-s + (0.0519 + 0.0299i)27-s + (−1.62 + 0.285i)29-s + (0.736 − 0.425i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.168 + 0.985i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.168 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.168 + 0.985i$
Analytic conductor: \(7.85611\)
Root analytic conductor: \(2.80287\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :2),\ -0.168 + 0.985i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.153841 - 0.182314i\)
\(L(\frac12)\) \(\approx\) \(0.153841 - 0.182314i\)
\(L(3)\) 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 \( 1 + (-113. + 342. i)T \)
good3 \( 1 + (8.26 - 9.85i)T + (-14.0 - 79.7i)T^{2} \)
5 \( 1 + (19.2 - 6.99i)T + (478. - 401. i)T^{2} \)
7 \( 1 + (-1.56 - 2.70i)T + (-1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (-57.2 + 99.1i)T + (-7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (73.1 + 87.2i)T + (-4.95e3 + 2.81e4i)T^{2} \)
17 \( 1 + (-11.6 + 66.0i)T + (-7.84e4 - 2.85e4i)T^{2} \)
23 \( 1 + (229. + 83.6i)T + (2.14e5 + 1.79e5i)T^{2} \)
29 \( 1 + (1.36e3 - 240. i)T + (6.64e5 - 2.41e5i)T^{2} \)
31 \( 1 + (-708. + 408. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 - 1.09e3iT - 1.87e6T^{2} \)
41 \( 1 + (51.0 - 60.7i)T + (-4.90e5 - 2.78e6i)T^{2} \)
43 \( 1 + (317. - 115. i)T + (2.61e6 - 2.19e6i)T^{2} \)
47 \( 1 + (275. + 1.56e3i)T + (-4.58e6 + 1.66e6i)T^{2} \)
53 \( 1 + (876. - 2.40e3i)T + (-6.04e6 - 5.07e6i)T^{2} \)
59 \( 1 + (4.80e3 + 846. i)T + (1.13e7 + 4.14e6i)T^{2} \)
61 \( 1 + (1.50e3 + 547. i)T + (1.06e7 + 8.89e6i)T^{2} \)
67 \( 1 + (4.75e3 - 838. i)T + (1.89e7 - 6.89e6i)T^{2} \)
71 \( 1 + (-183. - 503. i)T + (-1.94e7 + 1.63e7i)T^{2} \)
73 \( 1 + (5.42e3 + 4.55e3i)T + (4.93e6 + 2.79e7i)T^{2} \)
79 \( 1 + (-7.49e3 + 8.93e3i)T + (-6.76e6 - 3.83e7i)T^{2} \)
83 \( 1 + (3.23e3 + 5.59e3i)T + (-2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 + (6.88e3 + 8.20e3i)T + (-1.08e7 + 6.17e7i)T^{2} \)
97 \( 1 + (4.87e3 + 859. i)T + (8.31e7 + 3.02e7i)T^{2} \)
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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

−13.48713686075401626122681111230, −11.86446520860836297219693492006, −11.30703014642589382791885309041, −10.31323637294070725462794973654, −9.122869772528241243184322869393, −7.57517772905742905873792798305, −5.98728350819507193335170501832, −4.74435160361962751571293013951, −3.43799579919880489205828286750, −0.13921049020288012262505893868, 1.61073226220381847107556114227, 4.19937973710361926022042009398, 5.81718561650137585387738085871, 7.07404304568324037075295620097, 7.918915855286533707593339106984, 9.604188993892306331041335548691, 11.16734571748625822243178108031, 12.12862587363078543375506049899, 12.50812915658563224874469066167, 13.90789082068526311431487352330

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