Properties

Label 2-76-1.1-c3-0-3
Degree $2$
Conductor $76$
Sign $-1$
Analytic cond. $4.48414$
Root an. cond. $2.11758$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5.37·3-s + 11.8·5-s − 26.4·7-s + 1.86·9-s − 49.8·11-s − 49.0·13-s − 63.7·15-s + 17.2·17-s − 19·19-s + 142.·21-s + 166.·23-s + 15.6·25-s + 135.·27-s − 109.·29-s − 273.·31-s + 267.·33-s − 314.·35-s + 167.·37-s + 263.·39-s + 15.1·41-s + 413.·43-s + 22.0·45-s − 161.·47-s + 358.·49-s − 92.5·51-s − 490.·53-s − 591.·55-s + ⋯
L(s)  = 1  − 1.03·3-s + 1.06·5-s − 1.43·7-s + 0.0689·9-s − 1.36·11-s − 1.04·13-s − 1.09·15-s + 0.245·17-s − 0.229·19-s + 1.47·21-s + 1.51·23-s + 0.125·25-s + 0.962·27-s − 0.699·29-s − 1.58·31-s + 1.41·33-s − 1.51·35-s + 0.742·37-s + 1.08·39-s + 0.0577·41-s + 1.46·43-s + 0.0731·45-s − 0.501·47-s + 1.04·49-s − 0.254·51-s − 1.27·53-s − 1.44·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-1$
Analytic conductor: \(4.48414\)
Root analytic conductor: \(2.11758\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 76,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + 19T \)
good3 \( 1 + 5.37T + 27T^{2} \)
5 \( 1 - 11.8T + 125T^{2} \)
7 \( 1 + 26.4T + 343T^{2} \)
11 \( 1 + 49.8T + 1.33e3T^{2} \)
13 \( 1 + 49.0T + 2.19e3T^{2} \)
17 \( 1 - 17.2T + 4.91e3T^{2} \)
23 \( 1 - 166.T + 1.21e4T^{2} \)
29 \( 1 + 109.T + 2.43e4T^{2} \)
31 \( 1 + 273.T + 2.97e4T^{2} \)
37 \( 1 - 167.T + 5.06e4T^{2} \)
41 \( 1 - 15.1T + 6.89e4T^{2} \)
43 \( 1 - 413.T + 7.95e4T^{2} \)
47 \( 1 + 161.T + 1.03e5T^{2} \)
53 \( 1 + 490.T + 1.48e5T^{2} \)
59 \( 1 + 335.T + 2.05e5T^{2} \)
61 \( 1 - 725.T + 2.26e5T^{2} \)
67 \( 1 - 497.T + 3.00e5T^{2} \)
71 \( 1 + 798.T + 3.57e5T^{2} \)
73 \( 1 + 311.T + 3.89e5T^{2} \)
79 \( 1 + 665.T + 4.93e5T^{2} \)
83 \( 1 + 372.T + 5.71e5T^{2} \)
89 \( 1 + 673.T + 7.04e5T^{2} \)
97 \( 1 + 960.T + 9.12e5T^{2} \)
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

−12.99159099534972308548603158684, −12.75174266183647683931669042549, −11.10944874209083542019704076214, −10.13738636923699982955297322701, −9.289819188237190567974266006637, −7.24869569877946835792619065648, −6.00316224674392827034502495593, −5.21601495993563621097175110718, −2.74949077236954508699012559256, 0, 2.74949077236954508699012559256, 5.21601495993563621097175110718, 6.00316224674392827034502495593, 7.24869569877946835792619065648, 9.289819188237190567974266006637, 10.13738636923699982955297322701, 11.10944874209083542019704076214, 12.75174266183647683931669042549, 12.99159099534972308548603158684

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