Properties

Label 1-76-76.75-r0-0-0
Degree $1$
Conductor $76$
Sign $1$
Analytic cond. $0.352942$
Root an. cond. $0.352942$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s − 7-s + 9-s − 11-s − 13-s + 15-s + 17-s − 21-s − 23-s + 25-s + 27-s − 29-s + 31-s − 33-s − 35-s − 37-s − 39-s − 41-s − 43-s + 45-s − 47-s + 49-s + 51-s − 53-s − 55-s + 59-s + ⋯
L(s)  = 1  + 3-s + 5-s − 7-s + 9-s − 11-s − 13-s + 15-s + 17-s − 21-s − 23-s + 25-s + 27-s − 29-s + 31-s − 33-s − 35-s − 37-s − 39-s − 41-s − 43-s + 45-s − 47-s + 49-s + 51-s − 53-s − 55-s + 59-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(0.352942\)
Root analytic conductor: \(0.352942\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{76} (75, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 76,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.287000630\)
\(L(\frac12)\) \(\approx\) \(1.287000630\)
\(L(1)\) \(\approx\) \(1.337249851\)
\(L(1)\) \(\approx\) \(1.337249851\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 - T \)
11 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 + T \)
23 \( 1 - T \)
29 \( 1 - T \)
31 \( 1 + T \)
37 \( 1 - T \)
41 \( 1 - T \)
43 \( 1 - T \)
47 \( 1 - T \)
53 \( 1 - T \)
59 \( 1 + T \)
61 \( 1 + T \)
67 \( 1 + T \)
71 \( 1 + T \)
73 \( 1 + T \)
79 \( 1 + T \)
83 \( 1 - T \)
89 \( 1 - T \)
97 \( 1 - T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−31.742411544925891920425159430940, −30.09238558617870596295572083519, −29.41486871134307180863819261945, −28.25400108729443433901288765115, −26.61491924449366982256966535766, −25.92231168250869541964468098425, −25.12479853680107620905462960975, −24.000727185523306388055498829581, −22.413981882951850770734071857, −21.38540608952082941863952658631, −20.436942974250486087195291827645, −19.23670270979561450701239235769, −18.307920228896712125168295192920, −16.82600648030593608040981992716, −15.606521894277928419526228415964, −14.35097234624181254879080448543, −13.364122447975648155484403913337, −12.468799648816489039569645571430, −10.08539409531236167027430699124, −9.727828837994235452232957984486, −8.166301389804727017420484256839, −6.80525558008000178501586865609, −5.25059192447677962991272377620, −3.30247173383803505013623446369, −2.149932773489239013639461398865, 2.149932773489239013639461398865, 3.30247173383803505013623446369, 5.25059192447677962991272377620, 6.80525558008000178501586865609, 8.166301389804727017420484256839, 9.727828837994235452232957984486, 10.08539409531236167027430699124, 12.468799648816489039569645571430, 13.364122447975648155484403913337, 14.35097234624181254879080448543, 15.606521894277928419526228415964, 16.82600648030593608040981992716, 18.307920228896712125168295192920, 19.23670270979561450701239235769, 20.436942974250486087195291827645, 21.38540608952082941863952658631, 22.413981882951850770734071857, 24.000727185523306388055498829581, 25.12479853680107620905462960975, 25.92231168250869541964468098425, 26.61491924449366982256966535766, 28.25400108729443433901288765115, 29.41486871134307180863819261945, 30.09238558617870596295572083519, 31.742411544925891920425159430940

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