Properties

Label 2-76-76.51-c1-0-0
Degree $2$
Conductor $76$
Sign $0.480 - 0.877i$
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  + (−1.32 + 0.496i)2-s + (−1.09 − 0.397i)3-s + (1.50 − 1.31i)4-s + (0.615 + 3.49i)5-s + (1.64 − 0.0159i)6-s + (3.53 + 2.04i)7-s + (−1.34 + 2.48i)8-s + (−1.26 − 1.06i)9-s + (−2.55 − 4.31i)10-s + (−0.260 + 0.150i)11-s + (−2.16 + 0.836i)12-s + (0.546 + 1.50i)13-s + (−5.69 − 0.947i)14-s + (0.715 − 4.05i)15-s + (0.540 − 3.96i)16-s + (3.58 − 3.00i)17-s + ⋯
L(s)  = 1  + (−0.936 + 0.351i)2-s + (−0.629 − 0.229i)3-s + (0.753 − 0.657i)4-s + (0.275 + 1.56i)5-s + (0.670 − 0.00653i)6-s + (1.33 + 0.771i)7-s + (−0.474 + 0.880i)8-s + (−0.421 − 0.354i)9-s + (−0.806 − 1.36i)10-s + (−0.0784 + 0.0452i)11-s + (−0.625 + 0.241i)12-s + (0.151 + 0.416i)13-s + (−1.52 − 0.253i)14-s + (0.184 − 1.04i)15-s + (0.135 − 0.990i)16-s + (0.869 − 0.729i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.480 - 0.877i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.480 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.480 - 0.877i$
Analytic conductor: \(0.606863\)
Root analytic conductor: \(0.779014\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1/2),\ 0.480 - 0.877i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.511866 + 0.303283i\)
\(L(\frac12)\) \(\approx\) \(0.511866 + 0.303283i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.32 - 0.496i)T \)
19 \( 1 + (3.34 + 2.78i)T \)
good3 \( 1 + (1.09 + 0.397i)T + (2.29 + 1.92i)T^{2} \)
5 \( 1 + (-0.615 - 3.49i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (-3.53 - 2.04i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.260 - 0.150i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.546 - 1.50i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (-3.58 + 3.00i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (3.07 + 0.541i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (-0.344 + 0.410i)T + (-5.03 - 28.5i)T^{2} \)
31 \( 1 + (-2.08 + 3.60i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.42iT - 37T^{2} \)
41 \( 1 + (-2.70 + 7.42i)T + (-31.4 - 26.3i)T^{2} \)
43 \( 1 + (-2.23 + 0.393i)T + (40.4 - 14.7i)T^{2} \)
47 \( 1 + (-2.11 + 2.51i)T + (-8.16 - 46.2i)T^{2} \)
53 \( 1 + (5.12 + 0.903i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (-4.17 + 3.50i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (0.594 - 3.37i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-5.29 - 4.44i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (0.743 + 4.21i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (8.50 + 3.09i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (-0.187 - 0.0681i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (12.9 + 7.44i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.26 - 6.21i)T + (-68.1 + 57.2i)T^{2} \)
97 \( 1 + (-3.90 - 4.64i)T + (-16.8 + 95.5i)T^{2} \)
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

−14.67085841562874470140863399213, −14.25665140647826707911869925829, −11.87357530773972502650264255315, −11.31542902508259953089720078976, −10.42056648546200346106191870492, −8.999695763622572520106344010473, −7.67029633149904204601833986581, −6.53003223251182967033548964078, −5.53602199187925282224681253543, −2.39613538419653416340175997019, 1.36571741206083680434576937405, 4.44300370383208330437670396051, 5.77541583861468224658877877713, 7.988543549982668123797875393846, 8.425853906043312313460920807770, 10.02037371520602501313922208579, 10.90688517092860690260222984116, 11.94215915113691320017959555952, 12.89179944170411528796060316313, 14.30514922416806513128040798620

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