Properties

Label 2-736-184.45-c0-0-1
Degree $2$
Conductor $736$
Sign $1$
Analytic cond. $0.367311$
Root an. cond. $0.606062$
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 + 23-s − 25-s + 2·31-s − 2·41-s − 2·47-s + 49-s − 2·71-s − 2·73-s + 81-s + ⋯
L(s)  = 1  + 9-s + 23-s − 25-s + 2·31-s − 2·41-s − 2·47-s + 49-s − 2·71-s − 2·73-s + 81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(736\)    =    \(2^{5} \cdot 23\)
Sign: $1$
Analytic conductor: \(0.367311\)
Root analytic conductor: \(0.606062\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{736} (689, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 736,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.020002189\)
\(L(\frac12)\) \(\approx\) \(1.020002189\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−10.34086173798069995063619178125, −9.931216927381267557684444965364, −8.883292447689940333433485431726, −7.989708336333445553697347993073, −7.07739489961477510139449925390, −6.30459773414254707138227077588, −5.07544489633443435787886019043, −4.22533179623485325350588340286, −3.01936773641627600927367458111, −1.53465876695582332224291126939, 1.53465876695582332224291126939, 3.01936773641627600927367458111, 4.22533179623485325350588340286, 5.07544489633443435787886019043, 6.30459773414254707138227077588, 7.07739489961477510139449925390, 7.989708336333445553697347993073, 8.883292447689940333433485431726, 9.931216927381267557684444965364, 10.34086173798069995063619178125

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