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Properties

Label 2-76-19.8-c4-0-4
Degree $2$
Conductor $76$
Sign $-0.999 + 0.0419i$
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  + (−4.59 − 2.65i)3-s + (−1.89 + 3.27i)5-s + 36.8·7-s + (−26.4 − 45.7i)9-s − 226.·11-s + (−177. + 102. i)13-s + (17.3 − 10.0i)15-s + (15.7 − 27.2i)17-s + (−322. + 162. i)19-s + (−169. − 97.8i)21-s + (−439. − 761. i)23-s + (305. + 528. i)25-s + 710. i·27-s + (203. − 117. i)29-s − 1.65e3i·31-s + ⋯
L(s)  = 1  + (−0.510 − 0.294i)3-s + (−0.0756 + 0.131i)5-s + 0.751·7-s + (−0.325 − 0.564i)9-s − 1.87·11-s + (−1.05 + 0.606i)13-s + (0.0773 − 0.0446i)15-s + (0.0544 − 0.0942i)17-s + (−0.892 + 0.450i)19-s + (−0.384 − 0.221i)21-s + (−0.831 − 1.44i)23-s + (0.488 + 0.846i)25-s + 0.974i·27-s + (0.241 − 0.139i)29-s − 1.71i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0419i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0419i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.999 + 0.0419i$
Analytic conductor: \(7.85611\)
Root analytic conductor: \(2.80287\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :2),\ -0.999 + 0.0419i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.00395023 - 0.188211i\)
\(L(\frac12)\) \(\approx\) \(0.00395023 - 0.188211i\)
\(L(3)\) 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 \( 1 + (322. - 162. i)T \)
good3 \( 1 + (4.59 + 2.65i)T + (40.5 + 70.1i)T^{2} \)
5 \( 1 + (1.89 - 3.27i)T + (-312.5 - 541. i)T^{2} \)
7 \( 1 - 36.8T + 2.40e3T^{2} \)
11 \( 1 + 226.T + 1.46e4T^{2} \)
13 \( 1 + (177. - 102. i)T + (1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 + (-15.7 + 27.2i)T + (-4.17e4 - 7.23e4i)T^{2} \)
23 \( 1 + (439. + 761. i)T + (-1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (-203. + 117. i)T + (3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + 1.65e3iT - 9.23e5T^{2} \)
37 \( 1 - 1.06e3iT - 1.87e6T^{2} \)
41 \( 1 + (-70.5 - 40.7i)T + (1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (-617. + 1.07e3i)T + (-1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (-1.14e3 - 1.99e3i)T + (-2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + (-2.87e3 + 1.65e3i)T + (3.94e6 - 6.83e6i)T^{2} \)
59 \( 1 + (1.36e3 + 786. i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (-1.68e3 - 2.92e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (-2.80e3 + 1.61e3i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + (7.88e3 + 4.55e3i)T + (1.27e7 + 2.20e7i)T^{2} \)
73 \( 1 + (3.14e3 - 5.45e3i)T + (-1.41e7 - 2.45e7i)T^{2} \)
79 \( 1 + (705. + 407. i)T + (1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + 5.75e3T + 4.74e7T^{2} \)
89 \( 1 + (-1.01e4 + 5.86e3i)T + (3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 + (8.99e3 + 5.19e3i)T + (4.42e7 + 7.66e7i)T^{2} \)
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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

−13.04588973718150880683345359273, −12.12602042387602870324118342183, −11.09421063941161321009661754072, −10.05298237731660267843525508078, −8.438180916171730097463604118282, −7.34974417209479190219600890509, −5.90168913984644233231826406504, −4.62850858480029929602170342370, −2.41449135668425443792974285515, −0.094428519613016418428952869287, 2.48503669814828396740199567582, 4.79360862225142982354994820584, 5.51259446266675447295745234306, 7.52555926425712971828612504660, 8.396832894591205289082754408279, 10.23847001228281317111503202520, 10.80079834059193257262651893991, 12.07225239542163157481908984112, 13.15152259181626281509055528882, 14.32722404445587499009379537527

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