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Properties

Label 2-592-148.147-c0-0-1
Degree $2$
Conductor $592$
Sign $1$
Analytic cond. $0.295446$
Root an. cond. $0.543549$
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  + 9-s + 25-s − 37-s − 2·41-s + 49-s − 2·53-s − 2·73-s + 81-s − 2·101-s + ⋯
L(s)  = 1  + 9-s + 25-s − 37-s − 2·41-s + 49-s − 2·53-s − 2·73-s + 81-s − 2·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $1$
Analytic conductor: \(0.295446\)
Root analytic conductor: \(0.543549\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{592} (591, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 592,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9655459551\)
\(L(\frac12)\) \(\approx\) \(0.9655459551\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
37 \( 1 + T \)
good3 \( ( 1 - T )( 1 + T ) \)
5 \( ( 1 - T )( 1 + T ) \)
7 \( ( 1 - T )( 1 + T ) \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( 1 + T^{2} \)
23 \( 1 + T^{2} \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( 1 + T^{2} \)
41 \( ( 1 + T )^{2} \)
43 \( 1 + T^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 + T )^{2} \)
59 \( 1 + T^{2} \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( ( 1 + T )^{2} \)
79 \( 1 + T^{2} \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−10.73469907898234030687897339429, −10.11673214398674927139207956299, −9.183295736365396776875889613974, −8.282886060113288067633639661146, −7.23708579823634646922323361768, −6.55541606240268644791165769604, −5.27974799475477787592495810705, −4.35368810752309773011319690446, −3.17421212798072560752013885682, −1.62002315414002743460831486079, 1.62002315414002743460831486079, 3.17421212798072560752013885682, 4.35368810752309773011319690446, 5.27974799475477787592495810705, 6.55541606240268644791165769604, 7.23708579823634646922323361768, 8.282886060113288067633639661146, 9.183295736365396776875889613974, 10.11673214398674927139207956299, 10.73469907898234030687897339429

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