Properties

Label 2-76-76.15-c1-0-0
Degree $2$
Conductor $76$
Sign $0.400 - 0.916i$
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  + (−1.06 − 0.927i)2-s + (0.361 + 2.04i)3-s + (0.278 + 1.98i)4-s + (−2.99 + 2.51i)5-s + (1.51 − 2.52i)6-s + (0.0108 + 0.00626i)7-s + (1.54 − 2.37i)8-s + (−1.25 + 0.455i)9-s + (5.53 + 0.0969i)10-s + (3.15 − 1.82i)11-s + (−3.95 + 1.28i)12-s + (3.04 + 0.536i)13-s + (−0.00576 − 0.0167i)14-s + (−6.24 − 5.23i)15-s + (−3.84 + 1.10i)16-s + (−2.82 − 1.02i)17-s + ⋯
L(s)  = 1  + (−0.754 − 0.656i)2-s + (0.208 + 1.18i)3-s + (0.139 + 0.990i)4-s + (−1.34 + 1.12i)5-s + (0.618 − 1.03i)6-s + (0.00410 + 0.00236i)7-s + (0.544 − 0.838i)8-s + (−0.417 + 0.151i)9-s + (1.75 + 0.0306i)10-s + (0.952 − 0.550i)11-s + (−1.14 + 0.371i)12-s + (0.844 + 0.148i)13-s + (−0.00154 − 0.00447i)14-s + (−1.61 − 1.35i)15-s + (−0.961 + 0.275i)16-s + (−0.686 − 0.249i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.400 - 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.400 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.400 - 0.916i$
Analytic conductor: \(0.606863\)
Root analytic conductor: \(0.779014\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1/2),\ 0.400 - 0.916i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.515613 + 0.337545i\)
\(L(\frac12)\) \(\approx\) \(0.515613 + 0.337545i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.06 + 0.927i)T \)
19 \( 1 + (-3.96 - 1.81i)T \)
good3 \( 1 + (-0.361 - 2.04i)T + (-2.81 + 1.02i)T^{2} \)
5 \( 1 + (2.99 - 2.51i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (-0.0108 - 0.00626i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3.15 + 1.82i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.04 - 0.536i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (2.82 + 1.02i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (2.50 - 2.97i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (0.126 + 0.347i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (-1.29 + 2.24i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.87iT - 37T^{2} \)
41 \( 1 + (-7.88 + 1.39i)T + (38.5 - 14.0i)T^{2} \)
43 \( 1 + (5.11 + 6.09i)T + (-7.46 + 42.3i)T^{2} \)
47 \( 1 + (2.68 + 7.37i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (0.993 - 1.18i)T + (-9.20 - 52.1i)T^{2} \)
59 \( 1 + (-3.93 - 1.43i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (2.45 + 2.05i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (8.76 - 3.18i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-9.25 + 7.76i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-0.801 - 4.54i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (0.148 + 0.844i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-4.31 - 2.49i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.86 - 0.858i)T + (83.6 + 30.4i)T^{2} \)
97 \( 1 + (1.35 - 3.70i)T + (-74.3 - 62.3i)T^{2} \)
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

−15.03625629868113861527474384168, −13.77222458349178731952327401992, −11.84152734156100456860501445012, −11.29284837721063440855197128332, −10.38209665121112581599987994380, −9.254325996194497550746373445431, −8.127624535764367727711468221547, −6.81657782272243952902492033605, −4.00901212134792904127181113476, −3.38973916220022140047962941495, 1.20184909762105019397197168795, 4.50313483586985032137055263256, 6.43165580582090097832045034354, 7.55516982146695724670579867562, 8.319651583787419691160732358305, 9.277241709142559674058066759900, 11.20310634757337556427386105587, 12.18804405842626148007318414498, 13.17627971028029015959518870003, 14.42813878355236440291819494024

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