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Properties

Label 2-76-4.3-c2-0-0
Degree $2$
Conductor $76$
Sign $-0.866 + 0.499i$
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  + (−1 + 1.73i)2-s + 3.04i·3-s + (−1.99 − 3.46i)4-s − 8.54·5-s + (−5.27 − 3.04i)6-s − 3.04i·7-s + 7.99·8-s − 0.274·9-s + (8.54 − 14.8i)10-s + 13.0i·11-s + (10.5 − 6.09i)12-s − 21.8·13-s + (5.27 + 3.04i)14-s − 26.0i·15-s + (−8 + 13.8i)16-s − 7.27·17-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + 1.01i·3-s + (−0.499 − 0.866i)4-s − 1.70·5-s + (−0.879 − 0.507i)6-s − 0.435i·7-s + 0.999·8-s − 0.0305·9-s + (0.854 − 1.48i)10-s + 1.18i·11-s + (0.879 − 0.507i)12-s − 1.67·13-s + (0.376 + 0.217i)14-s − 1.73i·15-s + (−0.5 + 0.866i)16-s − 0.427·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.866 + 0.499i$
Analytic conductor: \(2.07085\)
Root analytic conductor: \(1.43904\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1),\ -0.866 + 0.499i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0985210 - 0.367685i\)
\(L(\frac12)\) \(\approx\) \(0.0985210 - 0.367685i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - 1.73i)T \)
19 \( 1 - 4.35iT \)
good3 \( 1 - 3.04iT - 9T^{2} \)
5 \( 1 + 8.54T + 25T^{2} \)
7 \( 1 + 3.04iT - 49T^{2} \)
11 \( 1 - 13.0iT - 121T^{2} \)
13 \( 1 + 21.8T + 169T^{2} \)
17 \( 1 + 7.27T + 289T^{2} \)
23 \( 1 - 31.8iT - 529T^{2} \)
29 \( 1 - 4.37T + 841T^{2} \)
31 \( 1 + 21.0iT - 961T^{2} \)
37 \( 1 - 24.1T + 1.36e3T^{2} \)
41 \( 1 - 6.74T + 1.68e3T^{2} \)
43 \( 1 + 14.0iT - 1.84e3T^{2} \)
47 \( 1 - 59.0iT - 2.20e3T^{2} \)
53 \( 1 + 26.9T + 2.80e3T^{2} \)
59 \( 1 + 76.7iT - 3.48e3T^{2} \)
61 \( 1 + 16.9T + 3.72e3T^{2} \)
67 \( 1 - 31.2iT - 4.48e3T^{2} \)
71 \( 1 + 25.4iT - 5.04e3T^{2} \)
73 \( 1 + 110.T + 5.32e3T^{2} \)
79 \( 1 - 104. iT - 6.24e3T^{2} \)
83 \( 1 + 0.376iT - 6.88e3T^{2} \)
89 \( 1 + 47.8T + 7.92e3T^{2} \)
97 \( 1 - 93.6T + 9.40e3T^{2} \)
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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

−15.26614631951368814647026235484, −14.49653989167092637913305017845, −12.66941284136967902796692738594, −11.37034496075491271213367983469, −10.15156645051961577363883938392, −9.345268806069075198547626287937, −7.71010912006590510594584597752, −7.21418394368749186378275950445, −4.85240841450964680616579909879, −4.12567214299558682290006844227, 0.38075786261024034835814460205, 2.76573626904382258291628180877, 4.46101652016326801070332098062, 7.01629702535989184296324053586, 7.923104400566152651706234610668, 8.807327855303875482613790456286, 10.55992759059793189461909514236, 11.77327076754771654675889425180, 12.18778460389666800233172683335, 13.14455781933853989774199308146

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