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Properties

Label 2-76-19.7-c3-0-3
Degree $2$
Conductor $76$
Sign $-0.350 + 0.936i$
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  + (1.07 + 1.86i)3-s + (−10.6 − 18.4i)5-s − 18.8·7-s + (11.1 − 19.3i)9-s + 8.41·11-s + (12.3 − 21.4i)13-s + (22.8 − 39.6i)15-s + (−18.3 − 31.7i)17-s + (−76.9 + 30.5i)19-s + (−20.2 − 35.1i)21-s + (19.6 − 34.0i)23-s + (−163. + 283. i)25-s + 106.·27-s + (57.8 − 100. i)29-s + 304.·31-s + ⋯
L(s)  = 1  + (0.207 + 0.358i)3-s + (−0.950 − 1.64i)5-s − 1.01·7-s + (0.414 − 0.717i)9-s + 0.230·11-s + (0.264 − 0.457i)13-s + (0.394 − 0.682i)15-s + (−0.261 − 0.452i)17-s + (−0.929 + 0.369i)19-s + (−0.210 − 0.365i)21-s + (0.178 − 0.308i)23-s + (−1.30 + 2.26i)25-s + 0.757·27-s + (0.370 − 0.642i)29-s + 1.76·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.350 + 0.936i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.350 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.350 + 0.936i$
Analytic conductor: \(4.48414\)
Root analytic conductor: \(2.11758\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :3/2),\ -0.350 + 0.936i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.547128 - 0.789118i\)
\(L(\frac12)\) \(\approx\) \(0.547128 - 0.789118i\)
\(L(\frac{5}{2})\) 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 \( 1 + (76.9 - 30.5i)T \)
good3 \( 1 + (-1.07 - 1.86i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (10.6 + 18.4i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + 18.8T + 343T^{2} \)
11 \( 1 - 8.41T + 1.33e3T^{2} \)
13 \( 1 + (-12.3 + 21.4i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (18.3 + 31.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
23 \( 1 + (-19.6 + 34.0i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-57.8 + 100. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 304.T + 2.97e4T^{2} \)
37 \( 1 + 265.T + 5.06e4T^{2} \)
41 \( 1 + (-172. - 299. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (108. + 187. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (47.7 - 82.7i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-347. + 601. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-221. - 384. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-188. + 327. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-412. + 714. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (207. + 358. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (66.0 + 114. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (349. + 605. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 1.35e3T + 5.71e5T^{2} \)
89 \( 1 + (56.8 - 98.4i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-207. - 359. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−13.36894756261868775712816359189, −12.56770975420196548062436382120, −11.81719331465351958564687072550, −10.06047852645765847022157514593, −9.049932466560510957857855986968, −8.185026555979857801859130782558, −6.48807538841823306681311919071, −4.71190935658767693489432560943, −3.62659562078917518057802276053, −0.60271819212407974061567620078, 2.66253091320932121468141475933, 4.01331125702110887204442097023, 6.51463926121772038012649632073, 7.10469176615125807101278388557, 8.428801840290306283225260015380, 10.16684229748022747074277830739, 10.93983224187604703637934372570, 12.13444457423147371298278298199, 13.36556624486636174054654293290, 14.33216931297269536902640554715

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