Properties

Label 2-76-19.16-c3-0-2
Degree $2$
Conductor $76$
Sign $0.852 - 0.522i$
Analytic cond. $4.48414$
Root an. cond. $2.11758$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.87 − 2.13i)3-s + (−3.07 + 17.4i)5-s + (12.3 + 21.4i)7-s + (9.25 − 7.76i)9-s + (35.0 − 60.6i)11-s + (−19.4 − 7.08i)13-s + (19.2 + 108. i)15-s + (−3.03 − 2.54i)17-s + (−0.940 − 82.8i)19-s + (118. + 99.4i)21-s + (18.9 + 107. i)23-s + (−176. − 64.2i)25-s + (−46.6 + 80.7i)27-s + (80.6 − 67.7i)29-s + (−126. − 218. i)31-s + ⋯
L(s)  = 1  + (1.13 − 0.411i)3-s + (−0.274 + 1.55i)5-s + (0.668 + 1.15i)7-s + (0.342 − 0.287i)9-s + (0.959 − 1.66i)11-s + (−0.415 − 0.151i)13-s + (0.330 + 1.87i)15-s + (−0.0433 − 0.0363i)17-s + (−0.0113 − 0.999i)19-s + (1.23 + 1.03i)21-s + (0.171 + 0.972i)23-s + (−1.41 − 0.514i)25-s + (−0.332 + 0.575i)27-s + (0.516 − 0.433i)29-s + (−0.730 − 1.26i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.852 - 0.522i$
Analytic conductor: \(4.48414\)
Root analytic conductor: \(2.11758\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :3/2),\ 0.852 - 0.522i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.99209 + 0.562177i\)
\(L(\frac12)\) \(\approx\) \(1.99209 + 0.562177i\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.940 + 82.8i)T \)
good3 \( 1 + (-5.87 + 2.13i)T + (20.6 - 17.3i)T^{2} \)
5 \( 1 + (3.07 - 17.4i)T + (-117. - 42.7i)T^{2} \)
7 \( 1 + (-12.3 - 21.4i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-35.0 + 60.6i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (19.4 + 7.08i)T + (1.68e3 + 1.41e3i)T^{2} \)
17 \( 1 + (3.03 + 2.54i)T + (853. + 4.83e3i)T^{2} \)
23 \( 1 + (-18.9 - 107. i)T + (-1.14e4 + 4.16e3i)T^{2} \)
29 \( 1 + (-80.6 + 67.7i)T + (4.23e3 - 2.40e4i)T^{2} \)
31 \( 1 + (126. + 218. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 200.T + 5.06e4T^{2} \)
41 \( 1 + (-391. + 142. i)T + (5.27e4 - 4.43e4i)T^{2} \)
43 \( 1 + (8.35 - 47.3i)T + (-7.47e4 - 2.71e4i)T^{2} \)
47 \( 1 + (199. - 167. i)T + (1.80e4 - 1.02e5i)T^{2} \)
53 \( 1 + (72.6 + 411. i)T + (-1.39e5 + 5.09e4i)T^{2} \)
59 \( 1 + (-418. - 351. i)T + (3.56e4 + 2.02e5i)T^{2} \)
61 \( 1 + (27.7 + 157. i)T + (-2.13e5 + 7.76e4i)T^{2} \)
67 \( 1 + (-37.8 + 31.7i)T + (5.22e4 - 2.96e5i)T^{2} \)
71 \( 1 + (-83.4 + 473. i)T + (-3.36e5 - 1.22e5i)T^{2} \)
73 \( 1 + (734. - 267. i)T + (2.98e5 - 2.50e5i)T^{2} \)
79 \( 1 + (707. - 257. i)T + (3.77e5 - 3.16e5i)T^{2} \)
83 \( 1 + (167. + 290. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-367. - 133. i)T + (5.40e5 + 4.53e5i)T^{2} \)
97 \( 1 + (-449. - 377. i)T + (1.58e5 + 8.98e5i)T^{2} \)
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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

−14.35148071073558469532037757875, −13.39250468300941933645277316824, −11.62884173410415216656729031755, −11.10536121752451007009224605668, −9.281649331043805185694880377030, −8.354409186510859223603717770133, −7.25738675693556553530943408474, −5.88180178875542203383363703396, −3.39633664509193602380381854658, −2.40023439046853997833255271201, 1.52010678709601315901788410946, 4.04218617055262860164946801122, 4.71767781813956397880847537827, 7.23491701130056302306327022769, 8.345929708685482300859798556824, 9.221721979492819574347971853008, 10.23994199404901095364357271158, 12.02542530811860156727859072032, 12.80781398147495323415982483048, 14.25465383951919224877977078599

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