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Properties

Label 2-48e2-4.3-c0-0-2
Degree $2$
Conductor $2304$
Sign $1$
Analytic cond. $1.14984$
Root an. cond. $1.07230$
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  + 2·17-s − 25-s + 2·41-s + 49-s + 2·73-s − 2·89-s − 2·97-s − 2·113-s + ⋯
L(s)  = 1  + 2·17-s − 25-s + 2·41-s + 49-s + 2·73-s − 2·89-s − 2·97-s − 2·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2304} (1279, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.248173258\)
\(L(\frac12)\) \(\approx\) \(1.248173258\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + T^{2} \)
7 \( ( 1 - T )( 1 + T ) \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( 1 + T^{2} \)
17 \( ( 1 - T )^{2} \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( 1 + T^{2} \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( 1 + T^{2} \)
41 \( ( 1 - T )^{2} \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( 1 + T^{2} \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( 1 + T^{2} \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( ( 1 - T )^{2} \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 + T )^{2} \)
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

−9.378673760287648319888056107075, −8.255332732991784687815928119209, −7.74543804295990927983572332868, −6.97250944904937553235091094934, −5.87509171124916124827712378705, −5.45123936390924824445883692033, −4.28045614835578633186481054865, −3.48193747071752939399822723581, −2.46333653315629702501640452945, −1.15387508176604843959225859948, 1.15387508176604843959225859948, 2.46333653315629702501640452945, 3.48193747071752939399822723581, 4.28045614835578633186481054865, 5.45123936390924824445883692033, 5.87509171124916124827712378705, 6.97250944904937553235091094934, 7.74543804295990927983572332868, 8.255332732991784687815928119209, 9.378673760287648319888056107075

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