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Properties

Label 2-76-19.9-c1-0-1
Degree $2$
Conductor $76$
Sign $0.840 + 0.541i$
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  + (1.86 − 1.56i)3-s + (−2.06 − 0.752i)5-s + (−1.48 + 2.57i)7-s + (0.509 − 2.89i)9-s + (1.34 + 2.33i)11-s + (3.64 + 3.05i)13-s + (−5.03 + 1.83i)15-s + (−1.19 − 6.79i)17-s + (−4.33 + 0.477i)19-s + (1.25 + 7.12i)21-s + (−4.86 + 1.77i)23-s + (−0.121 − 0.101i)25-s + (0.0792 + 0.137i)27-s + (1.17 − 6.63i)29-s + (2.14 − 3.71i)31-s + ⋯
L(s)  = 1  + (1.07 − 0.904i)3-s + (−0.924 − 0.336i)5-s + (−0.560 + 0.971i)7-s + (0.169 − 0.963i)9-s + (0.406 + 0.704i)11-s + (1.01 + 0.848i)13-s + (−1.30 + 0.473i)15-s + (−0.290 − 1.64i)17-s + (−0.993 + 0.109i)19-s + (0.274 + 1.55i)21-s + (−1.01 + 0.369i)23-s + (−0.0242 − 0.0203i)25-s + (0.0152 + 0.0264i)27-s + (0.217 − 1.23i)29-s + (0.384 − 0.666i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.840 + 0.541i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.840 + 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.840 + 0.541i$
Analytic conductor: \(0.606863\)
Root analytic conductor: \(0.779014\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1/2),\ 0.840 + 0.541i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03771 - 0.305438i\)
\(L(\frac12)\) \(\approx\) \(1.03771 - 0.305438i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.33 - 0.477i)T \)
good3 \( 1 + (-1.86 + 1.56i)T + (0.520 - 2.95i)T^{2} \)
5 \( 1 + (2.06 + 0.752i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (1.48 - 2.57i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.34 - 2.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.64 - 3.05i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (1.19 + 6.79i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (4.86 - 1.77i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-1.17 + 6.63i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-2.14 + 3.71i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.02T + 37T^{2} \)
41 \( 1 + (-1.42 + 1.19i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-8.50 - 3.09i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-0.108 + 0.617i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (6.42 - 2.33i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-0.623 - 3.53i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-9.43 + 3.43i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (1.58 - 8.96i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (0.533 + 0.194i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (0.598 - 0.502i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (-2.42 + 2.03i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-3.64 + 6.31i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.84 - 6.58i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (1.47 + 8.34i)T + (-91.1 + 33.1i)T^{2} \)
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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

−14.27519022709450320365428683077, −13.37325128919212044543280127446, −12.28569148126337437649122520299, −11.60232938896238708590829814521, −9.456277647325743244928594175813, −8.641390070452511158730736387942, −7.61328810229262724220872388146, −6.37460372075490149179782892675, −4.11255174678781203528980719732, −2.35201575711580772377083481448, 3.49696179161483934918036708375, 3.97100413618644408948050007232, 6.42424493897741305481972712634, 8.063235382847073726327715719598, 8.758721796620368334394631341730, 10.36594224214373633636747335916, 10.84452998669670605617900682655, 12.63493515030478167611413337633, 13.79177575048590821064396734061, 14.70403890910427482753776569486

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