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Properties

Label 2-76-19.7-c3-0-2
Degree $2$
Conductor $76$
Sign $0.946 + 0.321i$
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  + (−0.916 − 1.58i)3-s + (2.48 + 4.30i)5-s + 19.2·7-s + (11.8 − 20.4i)9-s + 27.9·11-s + (18.9 − 32.8i)13-s + (4.55 − 7.89i)15-s + (−1.97 − 3.41i)17-s + (32.9 + 75.9i)19-s + (−17.5 − 30.4i)21-s + (−41.8 + 72.5i)23-s + (50.1 − 86.8i)25-s − 92.7·27-s + (−60.6 + 105. i)29-s − 211.·31-s + ⋯
L(s)  = 1  + (−0.176 − 0.305i)3-s + (0.222 + 0.385i)5-s + 1.03·7-s + (0.437 − 0.758i)9-s + 0.766·11-s + (0.404 − 0.699i)13-s + (0.0784 − 0.135i)15-s + (−0.0281 − 0.0487i)17-s + (0.398 + 0.917i)19-s + (−0.182 − 0.316i)21-s + (−0.379 + 0.657i)23-s + (0.400 − 0.694i)25-s − 0.661·27-s + (−0.388 + 0.672i)29-s − 1.22·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 + 0.321i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.946 + 0.321i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.946 + 0.321i$
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: \((2,\ 76,\ (\ :3/2),\ 0.946 + 0.321i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.63096 - 0.269156i\)
\(L(\frac12)\) \(\approx\) \(1.63096 - 0.269156i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-32.9 - 75.9i)T \)
good3 \( 1 + (0.916 + 1.58i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (-2.48 - 4.30i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 - 19.2T + 343T^{2} \)
11 \( 1 - 27.9T + 1.33e3T^{2} \)
13 \( 1 + (-18.9 + 32.8i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (1.97 + 3.41i)T + (-2.45e3 + 4.25e3i)T^{2} \)
23 \( 1 + (41.8 - 72.5i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (60.6 - 105. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 211.T + 2.97e4T^{2} \)
37 \( 1 + 82.7T + 5.06e4T^{2} \)
41 \( 1 + (119. + 207. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-74.2 - 128. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (119. - 206. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-0.969 + 1.68i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-129. - 224. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-365. + 632. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-116. + 201. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-483. - 837. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (80.2 + 138. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (198. + 343. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 461.T + 5.71e5T^{2} \)
89 \( 1 + (552. - 956. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (391. + 678. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−14.15718135413203976603547709621, −12.75954432236328057545679642121, −11.78608388861604595150148342844, −10.74465544863726598548724543747, −9.467425473633080456149802145179, −8.090514251199126525194432099806, −6.85095450457542716455975794464, −5.55426056316293474352036757104, −3.74527135566097295678590137083, −1.46365236171090200579663452405, 1.70215416254781647065104034204, 4.24148246754445549178235752651, 5.32710430263313959423294791048, 7.04097266311283990927093917867, 8.429620190758801416223498218990, 9.519294987343593468446810778048, 10.91716498027301584299021028870, 11.65617638471066098447987475475, 13.09683546340295288869016601687, 14.06222640150956213197124648379

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