Properties

Degree $2$
Conductor $4624$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 4·7-s + 9-s + 6·11-s + 2·13-s + 4·19-s + 8·21-s − 5·25-s + 4·27-s − 4·31-s − 12·33-s + 4·37-s − 4·39-s − 6·41-s − 8·43-s + 9·49-s − 6·53-s − 8·57-s + 4·61-s − 4·63-s − 8·67-s − 2·73-s + 10·75-s − 24·77-s + 8·79-s − 11·81-s − 6·89-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.51·7-s + 1/3·9-s + 1.80·11-s + 0.554·13-s + 0.917·19-s + 1.74·21-s − 25-s + 0.769·27-s − 0.718·31-s − 2.08·33-s + 0.657·37-s − 0.640·39-s − 0.937·41-s − 1.21·43-s + 9/7·49-s − 0.824·53-s − 1.05·57-s + 0.512·61-s − 0.503·63-s − 0.977·67-s − 0.234·73-s + 1.15·75-s − 2.73·77-s + 0.900·79-s − 1.22·81-s − 0.635·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4624\)    =    \(2^{4} \cdot 17^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{4624} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4624,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9042577254\)
\(L(\frac12)\) \(\approx\) \(0.9042577254\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−17.85667446121155, −17.01423876808034, −16.75787143279406, −16.21712251941262, −15.68630110575037, −14.89374313752895, −14.09329209073494, −13.52736750889661, −12.81807124387314, −12.15842058538143, −11.64249902934558, −11.25819437962469, −10.29828915845897, −9.688527993269247, −9.218243206647949, −8.425390373429890, −7.248614481609208, −6.659295802666011, −6.136213663363790, −5.684914251973855, −4.635174832174788, −3.680769200687531, −3.207963622127117, −1.645703125590909, −0.5750481325221636, 0.5750481325221636, 1.645703125590909, 3.207963622127117, 3.680769200687531, 4.635174832174788, 5.684914251973855, 6.136213663363790, 6.659295802666011, 7.248614481609208, 8.425390373429890, 9.218243206647949, 9.688527993269247, 10.29828915845897, 11.25819437962469, 11.64249902934558, 12.15842058538143, 12.81807124387314, 13.52736750889661, 14.09329209073494, 14.89374313752895, 15.68630110575037, 16.21712251941262, 16.75787143279406, 17.01423876808034, 17.85667446121155

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