Properties

Label 2-76-19.16-c1-0-0
Degree $2$
Conductor $76$
Sign $-0.348 - 0.937i$
Analytic cond. $0.606863$
Root an. cond. $0.779014$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.83 + 1.03i)3-s + (−0.658 + 3.73i)5-s + (−0.0695 − 0.120i)7-s + (4.68 − 3.92i)9-s + (−0.350 + 0.607i)11-s + (−1.47 − 0.537i)13-s + (−1.98 − 11.2i)15-s + (2.88 + 2.41i)17-s + (4.28 + 0.790i)19-s + (0.321 + 0.269i)21-s + (1.02 + 5.82i)23-s + (−8.82 − 3.21i)25-s + (−4.69 + 8.13i)27-s + (5.28 − 4.43i)29-s + (1.43 + 2.49i)31-s + ⋯
L(s)  = 1  + (−1.63 + 0.596i)3-s + (−0.294 + 1.67i)5-s + (−0.0262 − 0.0455i)7-s + (1.56 − 1.30i)9-s + (−0.105 + 0.183i)11-s + (−0.409 − 0.149i)13-s + (−0.513 − 2.91i)15-s + (0.698 + 0.586i)17-s + (0.983 + 0.181i)19-s + (0.0701 + 0.0588i)21-s + (0.214 + 1.21i)23-s + (−1.76 − 0.642i)25-s + (−0.903 + 1.56i)27-s + (0.980 − 0.823i)29-s + (0.258 + 0.447i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.348 - 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.348 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.348 - 0.937i$
Analytic conductor: \(0.606863\)
Root analytic conductor: \(0.779014\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1/2),\ -0.348 - 0.937i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.299472 + 0.430989i\)
\(L(\frac12)\) \(\approx\) \(0.299472 + 0.430989i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.28 - 0.790i)T \)
good3 \( 1 + (2.83 - 1.03i)T + (2.29 - 1.92i)T^{2} \)
5 \( 1 + (0.658 - 3.73i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (0.0695 + 0.120i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.350 - 0.607i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.47 + 0.537i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-2.88 - 2.41i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (-1.02 - 5.82i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-5.28 + 4.43i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-1.43 - 2.49i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 6.33T + 37T^{2} \)
41 \( 1 + (4.40 - 1.60i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-0.935 + 5.30i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (1.42 - 1.19i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (-0.551 - 3.12i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (-1.81 - 1.52i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-0.587 - 3.33i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-7.61 + 6.38i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-0.375 + 2.12i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-10.6 + 3.89i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (-7.36 + 2.67i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (5.12 + 8.88i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (12.2 + 4.44i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (0.581 + 0.488i)T + (16.8 + 95.5i)T^{2} \)
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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

−15.18233710699236755645467530603, −13.92882344810257654291744113187, −12.14721323779112419859493978284, −11.50492497784583396921503558258, −10.45326361769469748294660195247, −9.956043507099343276853386715295, −7.47958696268780775201769076632, −6.46713191209666503375188205377, −5.29614662047458069582827100335, −3.54873994653399418692293757922, 0.900443169310256046672168622395, 4.76278256281295231663374637781, 5.46101280272107241771523204431, 6.97063644647217323837100314086, 8.328261117514410151275645689079, 9.809975838614096279218766404515, 11.25122937268747657943433617557, 12.25593017139625495779945522906, 12.55459560063759348721561669917, 13.82540512571743764713674541751

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