Properties

Label 2-26640-1.1-c1-0-2
Degree $2$
Conductor $26640$
Sign $1$
Analytic cond. $212.721$
Root an. cond. $14.5849$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 4·11-s − 2·13-s − 2·17-s − 4·19-s − 8·23-s + 25-s + 2·29-s − 8·31-s + 37-s − 10·41-s − 12·43-s − 7·49-s − 6·53-s − 4·55-s + 4·59-s − 10·61-s + 2·65-s + 4·67-s + 8·71-s − 6·73-s + 8·79-s − 4·83-s + 2·85-s − 10·89-s + 4·95-s + 10·97-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.20·11-s − 0.554·13-s − 0.485·17-s − 0.917·19-s − 1.66·23-s + 1/5·25-s + 0.371·29-s − 1.43·31-s + 0.164·37-s − 1.56·41-s − 1.82·43-s − 49-s − 0.824·53-s − 0.539·55-s + 0.520·59-s − 1.28·61-s + 0.248·65-s + 0.488·67-s + 0.949·71-s − 0.702·73-s + 0.900·79-s − 0.439·83-s + 0.216·85-s − 1.05·89-s + 0.410·95-s + 1.01·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(26640\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 37\)
Sign: $1$
Analytic conductor: \(212.721\)
Root analytic conductor: \(14.5849\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 26640,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9400581511\)
\(L(\frac12)\) \(\approx\) \(0.9400581511\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + T \)
37 \( 1 - T \)
good7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−15.33644820842580, −14.63605749523774, −14.38083064632247, −13.77173062271053, −13.08440432761635, −12.54469272186656, −12.05131210025955, −11.50840475905795, −11.16887566628753, −10.26083859447295, −9.953542769696813, −9.226101291860145, −8.657898961112167, −8.190393856264401, −7.541264145429681, −6.802666091616916, −6.452319753538562, −5.804065226503672, −4.848762810507059, −4.468690476288538, −3.662835424436557, −3.278385034942302, −2.013053213817352, −1.735868914489898, −0.3637662119109485, 0.3637662119109485, 1.735868914489898, 2.013053213817352, 3.278385034942302, 3.662835424436557, 4.468690476288538, 4.848762810507059, 5.804065226503672, 6.452319753538562, 6.802666091616916, 7.541264145429681, 8.190393856264401, 8.657898961112167, 9.226101291860145, 9.953542769696813, 10.26083859447295, 11.16887566628753, 11.50840475905795, 12.05131210025955, 12.54469272186656, 13.08440432761635, 13.77173062271053, 14.38083064632247, 14.63605749523774, 15.33644820842580

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