Properties

Label 2-76-19.8-c2-0-1
Degree $2$
Conductor $76$
Sign $0.796 + 0.604i$
Analytic cond. $2.07085$
Root an. cond. $1.43904$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 0.408i)3-s + (2.50 − 4.34i)5-s + 7.91·7-s + (−4.16 − 7.21i)9-s + 16.3·11-s + (−11.1 + 6.43i)13-s + (−3.55 + 2.05i)15-s + (−4.46 + 7.73i)17-s + (−18.5 − 4.20i)19-s + (−5.60 − 3.23i)21-s + (7.27 + 12.6i)23-s + (−0.0975 − 0.168i)25-s + 14.1i·27-s + (−29.7 + 17.1i)29-s + 25.5i·31-s + ⋯
L(s)  = 1  + (−0.235 − 0.136i)3-s + (0.501 − 0.869i)5-s + 1.13·7-s + (−0.462 − 0.801i)9-s + 1.48·11-s + (−0.857 + 0.495i)13-s + (−0.236 + 0.136i)15-s + (−0.262 + 0.455i)17-s + (−0.975 − 0.221i)19-s + (−0.266 − 0.154i)21-s + (0.316 + 0.548i)23-s + (−0.00390 − 0.00675i)25-s + 0.524i·27-s + (−1.02 + 0.592i)29-s + 0.823i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.796 + 0.604i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.796 + 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.796 + 0.604i$
Analytic conductor: \(2.07085\)
Root analytic conductor: \(1.43904\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1),\ 0.796 + 0.604i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.26108 - 0.423973i\)
\(L(\frac12)\) \(\approx\) \(1.26108 - 0.423973i\)
\(L(2)\) 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 \( 1 + (18.5 + 4.20i)T \)
good3 \( 1 + (0.707 + 0.408i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (-2.50 + 4.34i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 - 7.91T + 49T^{2} \)
11 \( 1 - 16.3T + 121T^{2} \)
13 \( 1 + (11.1 - 6.43i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (4.46 - 7.73i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + (-7.27 - 12.6i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (29.7 - 17.1i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 - 25.5iT - 961T^{2} \)
37 \( 1 + 49.2iT - 1.36e3T^{2} \)
41 \( 1 + (-27.8 - 16.0i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (4.51 - 7.81i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-19.9 - 34.5i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-73.6 + 42.5i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-53.9 - 31.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (38.1 + 66.1i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (69.8 - 40.3i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (11.0 + 6.37i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (42.2 - 73.2i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (106. + 61.6i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 21.9T + 6.88e3T^{2} \)
89 \( 1 + (98.0 - 56.6i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-146. - 84.5i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−14.42561625812013515672136105358, −12.91264949413776190893460045589, −11.97306728628732243308568409989, −11.09380631849497047690483959750, −9.313804453142100780524195667388, −8.728273798945884258685712102878, −6.99149043411491434929784675779, −5.60409961374469331738595602358, −4.26017759699514928575448480065, −1.54395315018668216234786232581, 2.29584901779094879969739897445, 4.51076751117253871137929725877, 5.95819999960680972104565533837, 7.31535273486782845959643126374, 8.660093607660894201663583251341, 10.12163924919760670302581042685, 11.08058134813302947503086685530, 11.90717462026378113011237495450, 13.55081139680935371644899819985, 14.54239991846361307201219527448

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