Properties

Label 2-76-19.11-c7-0-10
Degree $2$
Conductor $76$
Sign $-0.999 + 0.0213i$
Analytic cond. $23.7412$
Root an. cond. $4.87250$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (39.5 − 68.5i)3-s + (111. − 193. i)5-s − 546.·7-s + (−2.04e3 − 3.53e3i)9-s − 2.77e3·11-s + (−1.14e3 − 1.98e3i)13-s + (−8.82e3 − 1.52e4i)15-s + (1.25e4 − 2.17e4i)17-s + (−1.96e4 + 2.25e4i)19-s + (−2.16e4 + 3.74e4i)21-s + (1.48e4 + 2.56e4i)23-s + (1.42e4 + 2.46e4i)25-s − 1.50e5·27-s + (−4.46e4 − 7.73e4i)29-s + 1.64e5·31-s + ⋯
L(s)  = 1  + (0.846 − 1.46i)3-s + (0.398 − 0.690i)5-s − 0.602·7-s + (−0.934 − 1.61i)9-s − 0.628·11-s + (−0.144 − 0.250i)13-s + (−0.675 − 1.17i)15-s + (0.620 − 1.07i)17-s + (−0.655 + 0.755i)19-s + (−0.509 + 0.882i)21-s + (0.253 + 0.439i)23-s + (0.181 + 0.314i)25-s − 1.47·27-s + (−0.340 − 0.589i)29-s + 0.994·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0213i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0213i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.999 + 0.0213i$
Analytic conductor: \(23.7412\)
Root analytic conductor: \(4.87250\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :7/2),\ -0.999 + 0.0213i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.0200577 - 1.87542i\)
\(L(\frac12)\) \(\approx\) \(0.0200577 - 1.87542i\)
\(L(\frac{9}{2})\) 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 \( 1 + (1.96e4 - 2.25e4i)T \)
good3 \( 1 + (-39.5 + 68.5i)T + (-1.09e3 - 1.89e3i)T^{2} \)
5 \( 1 + (-111. + 193. i)T + (-3.90e4 - 6.76e4i)T^{2} \)
7 \( 1 + 546.T + 8.23e5T^{2} \)
11 \( 1 + 2.77e3T + 1.94e7T^{2} \)
13 \( 1 + (1.14e3 + 1.98e3i)T + (-3.13e7 + 5.43e7i)T^{2} \)
17 \( 1 + (-1.25e4 + 2.17e4i)T + (-2.05e8 - 3.55e8i)T^{2} \)
23 \( 1 + (-1.48e4 - 2.56e4i)T + (-1.70e9 + 2.94e9i)T^{2} \)
29 \( 1 + (4.46e4 + 7.73e4i)T + (-8.62e9 + 1.49e10i)T^{2} \)
31 \( 1 - 1.64e5T + 2.75e10T^{2} \)
37 \( 1 + 5.35e5T + 9.49e10T^{2} \)
41 \( 1 + (4.06e4 - 7.04e4i)T + (-9.73e10 - 1.68e11i)T^{2} \)
43 \( 1 + (-3.10e5 + 5.37e5i)T + (-1.35e11 - 2.35e11i)T^{2} \)
47 \( 1 + (3.19e5 + 5.53e5i)T + (-2.53e11 + 4.38e11i)T^{2} \)
53 \( 1 + (2.27e5 + 3.94e5i)T + (-5.87e11 + 1.01e12i)T^{2} \)
59 \( 1 + (-3.65e5 + 6.32e5i)T + (-1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (-4.45e5 - 7.71e5i)T + (-1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (-1.77e6 - 3.08e6i)T + (-3.03e12 + 5.24e12i)T^{2} \)
71 \( 1 + (-2.08e6 + 3.61e6i)T + (-4.54e12 - 7.87e12i)T^{2} \)
73 \( 1 + (1.65e6 - 2.86e6i)T + (-5.52e12 - 9.56e12i)T^{2} \)
79 \( 1 + (-1.76e6 + 3.06e6i)T + (-9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 - 7.47e6T + 2.71e13T^{2} \)
89 \( 1 + (2.65e6 + 4.59e6i)T + (-2.21e13 + 3.83e13i)T^{2} \)
97 \( 1 + (-7.83e6 + 1.35e7i)T + (-4.03e13 - 6.99e13i)T^{2} \)
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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

−12.79675140545806056663392261817, −11.92313268158127439502001658194, −10.00663546096184729491812042283, −8.856598171251019284028530624869, −7.86465557728901786361743043258, −6.80961625364940100169752926751, −5.42955509672084484565877434395, −3.18362940540379997822721406970, −1.89658671742509698087898705090, −0.54197616762041636311137844392, 2.50999834901504682101106491767, 3.51326186877025286553304775848, 4.88262206350997236303824507897, 6.47594608967298040960001927943, 8.194213466202162241269557593983, 9.296403052782176400501214527758, 10.26904481979378221233559395207, 10.81224294111001257331209372285, 12.70502507242768099360344361556, 13.95291711314738221560584958907

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