Properties

Label 2-76-1.1-c3-0-4
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.372·3-s − 16.8·5-s − 3.51·7-s − 26.8·9-s − 21.1·11-s + 14.0·13-s − 6.27·15-s − 17.2·17-s − 19·19-s − 1.30·21-s − 171.·23-s + 159.·25-s − 20.0·27-s + 264.·29-s + 185.·31-s − 7.86·33-s + 59.1·35-s + 212.·37-s + 5.24·39-s − 157.·41-s − 258.·43-s + 452.·45-s − 293.·47-s − 330.·49-s − 6.41·51-s + 215.·53-s + 356.·55-s + ⋯
L(s)  = 1  + 0.0716·3-s − 1.50·5-s − 0.189·7-s − 0.994·9-s − 0.579·11-s + 0.300·13-s − 0.108·15-s − 0.245·17-s − 0.229·19-s − 0.0135·21-s − 1.55·23-s + 1.27·25-s − 0.142·27-s + 1.69·29-s + 1.07·31-s − 0.0415·33-s + 0.285·35-s + 0.946·37-s + 0.0215·39-s − 0.598·41-s − 0.916·43-s + 1.50·45-s − 0.911·47-s − 0.964·49-s − 0.0176·51-s + 0.559·53-s + 0.873·55-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 \( 1 - 0.372T + 27T^{2} \)
5 \( 1 + 16.8T + 125T^{2} \)
7 \( 1 + 3.51T + 343T^{2} \)
11 \( 1 + 21.1T + 1.33e3T^{2} \)
13 \( 1 - 14.0T + 2.19e3T^{2} \)
17 \( 1 + 17.2T + 4.91e3T^{2} \)
23 \( 1 + 171.T + 1.21e4T^{2} \)
29 \( 1 - 264.T + 2.43e4T^{2} \)
31 \( 1 - 185.T + 2.97e4T^{2} \)
37 \( 1 - 212.T + 5.06e4T^{2} \)
41 \( 1 + 157.T + 6.89e4T^{2} \)
43 \( 1 + 258.T + 7.95e4T^{2} \)
47 \( 1 + 293.T + 1.03e5T^{2} \)
53 \( 1 - 215.T + 1.48e5T^{2} \)
59 \( 1 + 537.T + 2.05e5T^{2} \)
61 \( 1 + 280.T + 2.26e5T^{2} \)
67 \( 1 - 147.T + 3.00e5T^{2} \)
71 \( 1 + 913.T + 3.57e5T^{2} \)
73 \( 1 + 678.T + 3.89e5T^{2} \)
79 \( 1 + 608.T + 4.93e5T^{2} \)
83 \( 1 - 282.T + 5.71e5T^{2} \)
89 \( 1 + 214.T + 7.04e5T^{2} \)
97 \( 1 - 1.67e3T + 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.49788503722776660302967981564, −12.11275575634791519633455987561, −11.48177925080151079000655296282, −10.24642379119172767704449692881, −8.511644614083163592386551975397, −7.896494004176568198328012953839, −6.30155090606266999785009557310, −4.52575892854559574122108446838, −3.07106717455804559315000004384, 0, 3.07106717455804559315000004384, 4.52575892854559574122108446838, 6.30155090606266999785009557310, 7.896494004176568198328012953839, 8.511644614083163592386551975397, 10.24642379119172767704449692881, 11.48177925080151079000655296282, 12.11275575634791519633455987561, 13.49788503722776660302967981564

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