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Properties

Label 2-76-4.3-c4-0-17
Degree $2$
Conductor $76$
Sign $0.717 - 0.696i$
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  + (3.68 + 1.55i)2-s − 3.99i·3-s + (11.1 + 11.4i)4-s + 2.34·5-s + (6.22 − 14.7i)6-s + 66.6i·7-s + (23.1 + 59.6i)8-s + 65.0·9-s + (8.63 + 3.65i)10-s − 99.1i·11-s + (45.8 − 44.4i)12-s + 271.·13-s + (−103. + 245. i)14-s − 9.36i·15-s + (−7.78 + 255. i)16-s − 293.·17-s + ⋯
L(s)  = 1  + (0.920 + 0.389i)2-s − 0.443i·3-s + (0.696 + 0.717i)4-s + 0.0937·5-s + (0.172 − 0.408i)6-s + 1.35i·7-s + (0.361 + 0.932i)8-s + 0.803·9-s + (0.0863 + 0.0365i)10-s − 0.819i·11-s + (0.318 − 0.308i)12-s + 1.60·13-s + (−0.529 + 1.25i)14-s − 0.0416i·15-s + (−0.0304 + 0.999i)16-s − 1.01·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.717 - 0.696i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.717 - 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.717 - 0.696i$
Analytic conductor: \(7.85611\)
Root analytic conductor: \(2.80287\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :2),\ 0.717 - 0.696i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.78314 + 1.12809i\)
\(L(\frac12)\) \(\approx\) \(2.78314 + 1.12809i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.68 - 1.55i)T \)
19 \( 1 + 82.8iT \)
good3 \( 1 + 3.99iT - 81T^{2} \)
5 \( 1 - 2.34T + 625T^{2} \)
7 \( 1 - 66.6iT - 2.40e3T^{2} \)
11 \( 1 + 99.1iT - 1.46e4T^{2} \)
13 \( 1 - 271.T + 2.85e4T^{2} \)
17 \( 1 + 293.T + 8.35e4T^{2} \)
23 \( 1 - 136. iT - 2.79e5T^{2} \)
29 \( 1 + 1.23e3T + 7.07e5T^{2} \)
31 \( 1 + 957. iT - 9.23e5T^{2} \)
37 \( 1 - 2.22e3T + 1.87e6T^{2} \)
41 \( 1 + 1.06e3T + 2.82e6T^{2} \)
43 \( 1 + 866. iT - 3.41e6T^{2} \)
47 \( 1 + 4.23e3iT - 4.87e6T^{2} \)
53 \( 1 + 382.T + 7.89e6T^{2} \)
59 \( 1 - 4.30e3iT - 1.21e7T^{2} \)
61 \( 1 + 1.50e3T + 1.38e7T^{2} \)
67 \( 1 + 8.27e3iT - 2.01e7T^{2} \)
71 \( 1 + 1.01e3iT - 2.54e7T^{2} \)
73 \( 1 - 1.23e3T + 2.83e7T^{2} \)
79 \( 1 - 1.03e4iT - 3.89e7T^{2} \)
83 \( 1 - 2.00e3iT - 4.74e7T^{2} \)
89 \( 1 - 8.05e3T + 6.27e7T^{2} \)
97 \( 1 - 5.22e3T + 8.85e7T^{2} \)
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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

−13.47330593089383320654370797291, −13.22555713030627751227128270682, −11.87449614258181034940280240029, −11.08226684621395179076783626657, −9.059280820628849662513662616050, −7.961738392671236340933894009702, −6.43600479303341795867457946223, −5.64075056014056061804043920343, −3.85074331111721610239789205694, −2.09311432605571309972732633016, 1.47398265820660890793181306922, 3.77648816120286713045385918954, 4.52158272193952549516145684926, 6.31917521645647927039991183661, 7.46786122479330132099782035456, 9.548387694995380684695878806581, 10.54798442435876313482780825737, 11.26468057715692821466694197847, 12.92182994705253239969085597847, 13.41925058608002126215216847719

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