↑

Properties

Label 2-77-77.62-c1-0-5
Degree $2$
Conductor $77$
Sign $0.513 + 0.857i$
Analytic cond. $0.614848$
Root an. cond. $0.784122$
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  + (1.03 − 1.43i)2-s + (−0.348 − 1.07i)4-s + (−2.51 + 0.817i)7-s + (1.46 + 0.476i)8-s + (−2.42 − 1.76i)9-s + (−0.0629 + 3.31i)11-s + (−1.44 + 4.45i)14-s + (4.03 − 2.92i)16-s + (−5.04 + 1.63i)18-s + (4.67 + 3.53i)22-s + 2.56·23-s + (1.54 − 4.75i)25-s + (1.75 + 2.41i)28-s + (−7.02 + 2.28i)29-s − 5.72i·32-s + ⋯
L(s)  = 1  + (0.735 − 1.01i)2-s + (−0.174 − 0.536i)4-s + (−0.951 + 0.309i)7-s + (0.518 + 0.168i)8-s + (−0.809 − 0.587i)9-s + (−0.0189 + 0.999i)11-s + (−0.386 + 1.18i)14-s + (1.00 − 0.732i)16-s + (−1.18 + 0.386i)18-s + (0.997 + 0.754i)22-s + 0.533·23-s + (0.309 − 0.951i)25-s + (0.331 + 0.456i)28-s + (−1.30 + 0.423i)29-s − 1.01i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 + 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.513 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $0.513 + 0.857i$
Analytic conductor: \(0.614848\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 77,\ (\ :1/2),\ 0.513 + 0.857i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07590 - 0.609592i\)
\(L(\frac12)\) \(\approx\) \(1.07590 - 0.609592i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.51 - 0.817i)T \)
11 \( 1 + (0.0629 - 3.31i)T \)
good2 \( 1 + (-1.03 + 1.43i)T + (-0.618 - 1.90i)T^{2} \)
3 \( 1 + (2.42 + 1.76i)T^{2} \)
5 \( 1 + (-1.54 + 4.75i)T^{2} \)
13 \( 1 + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 2.56T + 23T^{2} \)
29 \( 1 + (7.02 - 2.28i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (3.68 + 11.3i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 2.77iT - 43T^{2} \)
47 \( 1 + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-11.5 - 8.41i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 12.5T + 67T^{2} \)
71 \( 1 + (12.9 - 9.43i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (10.3 - 14.1i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + (-29.9 - 92.2i)T^{2} \)
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

−14.10952473240440264594916999953, −12.83496210404122822679605629400, −12.35487537936776652893335292887, −11.27661089555602973467197247061, −10.09028318527243042282111084493, −8.941444054078672315956105000880, −7.14377303655170019470895158886, −5.57137144896879636119229479647, −3.92126450689799338293445164155, −2.58569582382856627395194458269, 3.44143414793870589705821960940, 5.23099003626911556878673020908, 6.22382220924576802860251920454, 7.39903797844460663265499732152, 8.730210530391355368258132373125, 10.28723014138534710875160056948, 11.43537956939211309199009897467, 13.14982682131617767900249354791, 13.58064875398064227331672669574, 14.66938901233728871005831468028

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