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Properties

Label 2-76-19.17-c1-0-1
Degree $2$
Conductor $76$
Sign $0.899 + 0.436i$
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  + (−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

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

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.899 + 0.436i$
Analytic conductor: \(0.606863\)
Root analytic conductor: \(0.779014\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1/2),\ 0.899 + 0.436i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.908797 - 0.208693i\)
\(L(\frac12)\) \(\approx\) \(0.908797 - 0.208693i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.84 + 3.94i)T \)
good3 \( 1 + (0.834 + 0.699i)T + (0.520 + 2.95i)T^{2} \)
5 \( 1 + (-3.00 + 1.09i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (-0.278 - 0.482i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.96 - 3.39i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.19 - 2.67i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (0.660 - 3.74i)T + (-15.9 - 5.81i)T^{2} \)
23 \( 1 + (-4.67 - 1.70i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (-0.0201 - 0.114i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (3.54 + 6.13i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 5.67T + 37T^{2} \)
41 \( 1 + (-9.20 - 7.72i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-6.74 + 2.45i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (0.00419 + 0.0237i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (8.18 + 2.97i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (-1.88 + 10.7i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (11.7 + 4.29i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-2.27 - 12.8i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (-3.17 + 1.15i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (-0.338 - 0.284i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (-1.31 - 1.10i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (2.90 + 5.02i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.83 + 1.53i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (-1.86 + 10.5i)T + (-91.1 - 33.1i)T^{2} \)
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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

−14.42132497675847109177316524668, −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 $Z$-function along the critical line