Properties

Label 2-76-19.11-c1-0-0
Degree $2$
Conductor $76$
Sign $0.910 + 0.412i$
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  + (0.5 − 0.866i)3-s + (0.5 − 0.866i)5-s + (1 + 1.73i)9-s − 4·11-s + (0.5 + 0.866i)13-s + (−0.499 − 0.866i)15-s + (−1.5 + 2.59i)17-s + (−4 + 1.73i)19-s + (−2.5 − 4.33i)23-s + (2 + 3.46i)25-s + 5·27-s + (−3.5 − 6.06i)29-s + 4·31-s + (−2 + 3.46i)33-s + 10·37-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (0.223 − 0.387i)5-s + (0.333 + 0.577i)9-s − 1.20·11-s + (0.138 + 0.240i)13-s + (−0.129 − 0.223i)15-s + (−0.363 + 0.630i)17-s + (−0.917 + 0.397i)19-s + (−0.521 − 0.902i)23-s + (0.400 + 0.692i)25-s + 0.962·27-s + (−0.649 − 1.12i)29-s + 0.718·31-s + (−0.348 + 0.603i)33-s + 1.64·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.910 + 0.412i$
Analytic conductor: \(0.606863\)
Root analytic conductor: \(0.779014\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1/2),\ 0.910 + 0.412i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.986847 - 0.213284i\)
\(L(\frac12)\) \(\approx\) \(0.986847 - 0.213284i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4 - 1.73i)T \)
good3 \( 1 + (-0.5 + 0.866i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (2.5 + 4.33i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.5 + 6.06i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 10T + 37T^{2} \)
41 \( 1 + (-2.5 + 4.33i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.5 + 4.33i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.5 - 6.06i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.5 + 9.52i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.5 + 9.52i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.5 - 2.59i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.5 - 9.52i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (7.5 - 12.9i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.5 + 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (1.5 + 2.59i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.5 + 4.33i)T + (-48.5 - 84.0i)T^{2} \)
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

−14.29974575719456045326622773158, −13.11069918964448992657991348226, −12.74655459434758243945139301072, −11.06774901692491150864147072101, −10.03822225854232995061329335650, −8.523628327780524062372942515050, −7.65381165678650911206172693551, −6.09883239151249176774240439136, −4.53039496134745697616897108822, −2.23275176588073520095478123304, 2.88959967239244464973068077936, 4.60432761726607850681992359706, 6.24816901613009100736364334993, 7.68782966426915268740231684369, 9.083872573304780836458674175901, 10.13794998726485266646247897559, 11.07130322013113521553593114138, 12.56425310513379079331226375170, 13.54349859961333264765282700419, 14.76586366114238299366274732007

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