↑

Properties

Label 2-76-19.9-c1-0-0
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

Related objects

Downloads

Learn more

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: $\chi_{76} (9, \cdot )$
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 \( 1 + (-1.86 - 10.5i)T + (-91.1 + 33.1i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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