Properties

Label 2-95-95.94-c0-0-1
Degree $2$
Conductor $95$
Sign $1$
Analytic cond. $0.0474111$
Root an. cond. $0.217741$
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  − 4-s + 5-s − 9-s − 2·11-s + 16-s + 19-s − 20-s + 25-s + 36-s + 2·44-s − 45-s + 49-s − 2·55-s − 2·61-s − 64-s − 76-s + 80-s + 81-s + 95-s + 2·99-s − 100-s − 2·101-s + ⋯
L(s)  = 1  − 4-s + 5-s − 9-s − 2·11-s + 16-s + 19-s − 20-s + 25-s + 36-s + 2·44-s − 45-s + 49-s − 2·55-s − 2·61-s − 64-s − 76-s + 80-s + 81-s + 95-s + 2·99-s − 100-s − 2·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $1$
Analytic conductor: \(0.0474111\)
Root analytic conductor: \(0.217741\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{95} (94, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 95,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4985172696\)
\(L(\frac12)\) \(\approx\) \(0.4985172696\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
19 \( 1 - T \)
good2 \( 1 + T^{2} \)
3 \( 1 + T^{2} \)
7 \( ( 1 - T )( 1 + T ) \)
11 \( ( 1 + T )^{2} \)
13 \( 1 + T^{2} \)
17 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( 1 + T^{2} \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( 1 + T^{2} \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 + T )^{2} \)
67 \( 1 + T^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( 1 + T^{2} \)
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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

−13.90530768021683048703931500792, −13.47265417168699841911242379081, −12.42493324594992440640465564801, −10.80601861381863672990663374102, −9.891081373460128606623069373424, −8.844029451101821452107687940364, −7.76012765378292796390012623669, −5.76418806130242862195870775024, −5.05764801474219645507661583018, −2.87350641693744207631912289409, 2.87350641693744207631912289409, 5.05764801474219645507661583018, 5.76418806130242862195870775024, 7.76012765378292796390012623669, 8.844029451101821452107687940364, 9.891081373460128606623069373424, 10.80601861381863672990663374102, 12.42493324594992440640465564801, 13.47265417168699841911242379081, 13.90530768021683048703931500792

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