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Properties

Label 2-295-295.294-c0-0-2
Degree $2$
Conductor $295$
Sign $1$
Analytic cond. $0.147224$
Root an. cond. $0.383698$
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 + 16-s − 2·19-s − 20-s + 25-s − 2·29-s − 36-s − 2·41-s + 45-s + 49-s + 59-s − 64-s − 2·71-s + 2·76-s + 2·79-s + 80-s + 81-s − 2·95-s − 100-s + 2·116-s + ⋯
L(s)  = 1  − 4-s + 5-s + 9-s + 16-s − 2·19-s − 20-s + 25-s − 2·29-s − 36-s − 2·41-s + 45-s + 49-s + 59-s − 64-s − 2·71-s + 2·76-s + 2·79-s + 80-s + 81-s − 2·95-s − 100-s + 2·116-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 295 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 295 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(295\)    =    \(5 \cdot 59\)
Sign: $1$
Analytic conductor: \(0.147224\)
Root analytic conductor: \(0.383698\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{295} (294, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 295,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7544110325\)
\(L(\frac12)\) \(\approx\) \(0.7544110325\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−12.30316135223849536748441314461, −10.73873661196391459901143738163, −10.06339346522991188767713250654, −9.233268245566741331454016685878, −8.423602640577405702990143688298, −7.08287576499635293686916466191, −5.95685201287422724050475808782, −4.87654148899374470308483011361, −3.83650161991916834317285358945, −1.88469320597866519526099075548, 1.88469320597866519526099075548, 3.83650161991916834317285358945, 4.87654148899374470308483011361, 5.95685201287422724050475808782, 7.08287576499635293686916466191, 8.423602640577405702990143688298, 9.233268245566741331454016685878, 10.06339346522991188767713250654, 10.73873661196391459901143738163, 12.30316135223849536748441314461

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