Properties

Label 2-76-76.15-c1-0-6
Degree $2$
Conductor $76$
Sign $0.869 + 0.493i$
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  + (1.41 − 0.0247i)2-s + (−0.361 − 2.04i)3-s + (1.99 − 0.0700i)4-s + (−2.99 + 2.51i)5-s + (−0.561 − 2.88i)6-s + (−0.0108 − 0.00626i)7-s + (2.82 − 0.148i)8-s + (−1.25 + 0.455i)9-s + (−4.17 + 3.63i)10-s + (−3.15 + 1.82i)11-s + (−0.865 − 4.07i)12-s + (3.04 + 0.536i)13-s + (−0.0155 − 0.00859i)14-s + (6.24 + 5.23i)15-s + (3.99 − 0.279i)16-s + (−2.82 − 1.02i)17-s + ⋯
L(s)  = 1  + (0.999 − 0.0175i)2-s + (−0.208 − 1.18i)3-s + (0.999 − 0.0350i)4-s + (−1.34 + 1.12i)5-s + (−0.229 − 1.17i)6-s + (−0.00410 − 0.00236i)7-s + (0.998 − 0.0525i)8-s + (−0.417 + 0.151i)9-s + (−1.32 + 1.14i)10-s + (−0.952 + 0.550i)11-s + (−0.249 − 1.17i)12-s + (0.844 + 0.148i)13-s + (−0.00414 − 0.00229i)14-s + (1.61 + 1.35i)15-s + (0.997 − 0.0699i)16-s + (−0.686 − 0.249i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 + 0.493i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.869 + 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.869 + 0.493i$
Analytic conductor: \(0.606863\)
Root analytic conductor: \(0.779014\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1/2),\ 0.869 + 0.493i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25360 - 0.330804i\)
\(L(\frac12)\) \(\approx\) \(1.25360 - 0.330804i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 + 0.0247i)T \)
19 \( 1 + (3.96 + 1.81i)T \)
good3 \( 1 + (0.361 + 2.04i)T + (-2.81 + 1.02i)T^{2} \)
5 \( 1 + (2.99 - 2.51i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (0.0108 + 0.00626i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.15 - 1.82i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.04 - 0.536i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (2.82 + 1.02i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (-2.50 + 2.97i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (0.126 + 0.347i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (1.29 - 2.24i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.87iT - 37T^{2} \)
41 \( 1 + (-7.88 + 1.39i)T + (38.5 - 14.0i)T^{2} \)
43 \( 1 + (-5.11 - 6.09i)T + (-7.46 + 42.3i)T^{2} \)
47 \( 1 + (-2.68 - 7.37i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (0.993 - 1.18i)T + (-9.20 - 52.1i)T^{2} \)
59 \( 1 + (3.93 + 1.43i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (2.45 + 2.05i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-8.76 + 3.18i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (9.25 - 7.76i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-0.801 - 4.54i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (-0.148 - 0.844i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (4.31 + 2.49i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.86 - 0.858i)T + (83.6 + 30.4i)T^{2} \)
97 \( 1 + (1.35 - 3.70i)T + (-74.3 - 62.3i)T^{2} \)
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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

−14.36795369473258059024557536760, −13.13970719093386759416579055838, −12.46622255531485063960825940068, −11.31873884597632677162527108023, −10.73433716422898530323304687172, −8.019909452794423725672543287390, −7.13069309390796452099414978487, −6.38203899090658716041133273751, −4.32394483908086047694959677991, −2.65147361011448055261926687937, 3.69448222703263475387544372482, 4.52029881054641722468705443005, 5.64877330549932787587582860885, 7.69925528532798444390143935989, 8.847858725047753759285535551553, 10.68453579365396204138292566508, 11.25687445856982925662820674877, 12.55599739829908430976605934580, 13.32974318829291344907142904085, 14.99722377771540882932615587447

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