Properties

Degree $2$
Conductor $94640$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 7-s − 3·9-s + 4·11-s + 2·17-s + 25-s + 6·29-s + 8·31-s − 35-s + 10·37-s − 2·41-s − 4·43-s − 3·45-s + 8·47-s + 49-s − 2·53-s + 4·55-s − 8·59-s − 14·61-s + 3·63-s − 12·67-s − 16·71-s − 2·73-s − 4·77-s + 8·79-s + 9·81-s + 8·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s − 9-s + 1.20·11-s + 0.485·17-s + 1/5·25-s + 1.11·29-s + 1.43·31-s − 0.169·35-s + 1.64·37-s − 0.312·41-s − 0.609·43-s − 0.447·45-s + 1.16·47-s + 1/7·49-s − 0.274·53-s + 0.539·55-s − 1.04·59-s − 1.79·61-s + 0.377·63-s − 1.46·67-s − 1.89·71-s − 0.234·73-s − 0.455·77-s + 0.900·79-s + 81-s + 0.878·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 94640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(94640\)    =    \(2^{4} \cdot 5 \cdot 7 \cdot 13^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{94640} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 94640,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
7 \( 1 + T \)
13 \( 1 \)
good3 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−14.01322697846882, −13.48467591017298, −13.42230388860616, −12.28434677682235, −12.12268115643043, −11.82797220042932, −11.01272912545965, −10.69518430444083, −9.999706355416090, −9.591865745368838, −9.075385442407640, −8.692302670788807, −8.061602260573181, −7.600098314006950, −6.785316896286436, −6.351309135801081, −5.997244641517107, −5.505876648829624, −4.568604910282922, −4.375976735786316, −3.429156166736373, −2.909461432773250, −2.520747568285015, −1.462357819627640, −1.014813132360112, 0, 1.014813132360112, 1.462357819627640, 2.520747568285015, 2.909461432773250, 3.429156166736373, 4.375976735786316, 4.568604910282922, 5.505876648829624, 5.997244641517107, 6.351309135801081, 6.785316896286436, 7.600098314006950, 8.061602260573181, 8.692302670788807, 9.075385442407640, 9.591865745368838, 9.999706355416090, 10.69518430444083, 11.01272912545965, 11.82797220042932, 12.12268115643043, 12.28434677682235, 13.42230388860616, 13.48467591017298, 14.01322697846882

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