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Properties

Label 2-1520-95.94-c0-0-2
Degree $2$
Conductor $1520$
Sign $1$
Analytic cond. $0.758578$
Root an. cond. $0.870964$
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  + 5-s − 9-s + 2·11-s − 19-s + 25-s − 45-s + 49-s + 2·55-s − 2·61-s + 81-s − 95-s − 2·99-s − 2·101-s + ⋯
L(s)  = 1  + 5-s − 9-s + 2·11-s − 19-s + 25-s − 45-s + 49-s + 2·55-s − 2·61-s + 81-s − 95-s − 2·99-s − 2·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $1$
Analytic conductor: \(0.758578\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1520} (1329, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1520,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.331803221\)
\(L(\frac12)\) \(\approx\) \(1.331803221\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
19 \( 1 + T \)
good3 \( 1 + T^{2} \)
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

−9.416010247014081163035057639074, −9.027606780314589063955463168702, −8.314827988416942982929500973221, −7.02068134836168526815844492735, −6.29215254638267265602568859164, −5.80012734247772914022913454480, −4.65499219945786503864110380295, −3.66507747376751859802743192347, −2.51739801830774741534626891070, −1.41974404229939219259928729519, 1.41974404229939219259928729519, 2.51739801830774741534626891070, 3.66507747376751859802743192347, 4.65499219945786503864110380295, 5.80012734247772914022913454480, 6.29215254638267265602568859164, 7.02068134836168526815844492735, 8.314827988416942982929500973221, 9.027606780314589063955463168702, 9.416010247014081163035057639074

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