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Properties

Label 2-76-19.17-c1-0-1
Degree 22
Conductor 7676
Sign 0.899+0.436i0.899 + 0.436i
Analytic cond. 0.6068630.606863
Root an. cond. 0.7790140.779014
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.834 − 0.699i)3-s + (3.00 − 1.09i)5-s + (0.278 + 0.482i)7-s + (−0.315 − 1.78i)9-s + (−1.96 + 3.39i)11-s + (−3.19 + 2.67i)13-s + (−3.27 − 1.19i)15-s + (−0.660 + 3.74i)17-s + (−1.84 − 3.94i)19-s + (0.105 − 0.597i)21-s + (4.67 + 1.70i)23-s + (4.01 − 3.36i)25-s + (−2.62 + 4.53i)27-s + (0.0201 + 0.114i)29-s + (−3.54 − 6.13i)31-s + ⋯
L(s)  = 1  + (−0.481 − 0.404i)3-s + (1.34 − 0.489i)5-s + (0.105 + 0.182i)7-s + (−0.105 − 0.595i)9-s + (−0.591 + 1.02i)11-s + (−0.885 + 0.743i)13-s + (−0.845 − 0.307i)15-s + (−0.160 + 0.907i)17-s + (−0.423 − 0.906i)19-s + (0.0229 − 0.130i)21-s + (0.974 + 0.354i)23-s + (0.803 − 0.673i)25-s + (−0.504 + 0.873i)27-s + (0.00374 + 0.0212i)29-s + (−0.636 − 1.10i)31-s + ⋯

Functional equation

Λ(s)=(76s/2 ΓC(s) L(s)=((0.899+0.436i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.436i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(76s/2 ΓC(s+1/2) L(s)=((0.899+0.436i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.899 + 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7676    =    22⋅192^{2} \cdot 19
Sign: 0.899+0.436i0.899 + 0.436i
Analytic conductor: 0.6068630.606863
Root analytic conductor: 0.7790140.779014
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ76(17,⋅)\chi_{76} (17, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 76, ( :1/2), 0.899+0.436i)(2,\ 76,\ (\ :1/2),\ 0.899 + 0.436i)

Particular Values

L(1)L(1) ≈\approx 0.908797−0.208693i0.908797 - 0.208693i
L(12)L(\frac12) ≈\approx 0.908797−0.208693i0.908797 - 0.208693i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
19 1+(1.84+3.94i)T 1 + (1.84 + 3.94i)T
good3 1+(0.834+0.699i)T+(0.520+2.95i)T2 1 + (0.834 + 0.699i)T + (0.520 + 2.95i)T^{2}
5 1+(−3.00+1.09i)T+(3.83−3.21i)T2 1 + (-3.00 + 1.09i)T + (3.83 - 3.21i)T^{2}
7 1+(−0.278−0.482i)T+(−3.5+6.06i)T2 1 + (-0.278 - 0.482i)T + (-3.5 + 6.06i)T^{2}
11 1+(1.96−3.39i)T+(−5.5−9.52i)T2 1 + (1.96 - 3.39i)T + (-5.5 - 9.52i)T^{2}
13 1+(3.19−2.67i)T+(2.25−12.8i)T2 1 + (3.19 - 2.67i)T + (2.25 - 12.8i)T^{2}
17 1+(0.660−3.74i)T+(−15.9−5.81i)T2 1 + (0.660 - 3.74i)T + (-15.9 - 5.81i)T^{2}
23 1+(−4.67−1.70i)T+(17.6+14.7i)T2 1 + (-4.67 - 1.70i)T + (17.6 + 14.7i)T^{2}
29 1+(−0.0201−0.114i)T+(−27.2+9.91i)T2 1 + (-0.0201 - 0.114i)T + (-27.2 + 9.91i)T^{2}
31 1+(3.54+6.13i)T+(−15.5+26.8i)T2 1 + (3.54 + 6.13i)T + (-15.5 + 26.8i)T^{2}
37 1+5.67T+37T2 1 + 5.67T + 37T^{2}
41 1+(−9.20−7.72i)T+(7.11+40.3i)T2 1 + (-9.20 - 7.72i)T + (7.11 + 40.3i)T^{2}
43 1+(−6.74+2.45i)T+(32.9−27.6i)T2 1 + (-6.74 + 2.45i)T + (32.9 - 27.6i)T^{2}
47 1+(0.00419+0.0237i)T+(−44.1+16.0i)T2 1 + (0.00419 + 0.0237i)T + (-44.1 + 16.0i)T^{2}
53 1+(8.18+2.97i)T+(40.6+34.0i)T2 1 + (8.18 + 2.97i)T + (40.6 + 34.0i)T^{2}
59 1+(−1.88+10.7i)T+(−55.4−20.1i)T2 1 + (-1.88 + 10.7i)T + (-55.4 - 20.1i)T^{2}
61 1+(11.7+4.29i)T+(46.7+39.2i)T2 1 + (11.7 + 4.29i)T + (46.7 + 39.2i)T^{2}
67 1+(−2.27−12.8i)T+(−62.9+22.9i)T2 1 + (-2.27 - 12.8i)T + (-62.9 + 22.9i)T^{2}
71 1+(−3.17+1.15i)T+(54.3−45.6i)T2 1 + (-3.17 + 1.15i)T + (54.3 - 45.6i)T^{2}
73 1+(−0.338−0.284i)T+(12.6+71.8i)T2 1 + (-0.338 - 0.284i)T + (12.6 + 71.8i)T^{2}
79 1+(−1.31−1.10i)T+(13.7+77.7i)T2 1 + (-1.31 - 1.10i)T + (13.7 + 77.7i)T^{2}
83 1+(2.90+5.02i)T+(−41.5+71.8i)T2 1 + (2.90 + 5.02i)T + (-41.5 + 71.8i)T^{2}
89 1+(−1.83+1.53i)T+(15.4−87.6i)T2 1 + (-1.83 + 1.53i)T + (15.4 - 87.6i)T^{2}
97 1+(−1.86+10.5i)T+(−91.1−33.1i)T2 1 + (-1.86 + 10.5i)T + (-91.1 - 33.1i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−14.42132497675847109177316524668, −13.01449916753309294567366676325, −12.60615347896638500951284125645, −11.21612627929494277208417656494, −9.796543658751649818778285741020, −9.051653752364956815312864369710, −7.18938297363131295390613072542, −6.02769541008150040947497117186, −4.83062161754843451185583484967, −2.05046261599182544128095747837, 2.67993320624799818480210094506, 5.09272918690391610210629142956, 5.94784578336753004732095096395, 7.55392246888851403107939255643, 9.174095304755864053013076302092, 10.52489651494538019896668980920, 10.76387517127708070766473489228, 12.52201386551504989283397396602, 13.72003487473720876579840719793, 14.34061974395676903792073792611

Graph of the ZZ-function along the critical line