Properties

Label 2-86640-1.1-c1-0-2
Degree $2$
Conductor $86640$
Sign $1$
Analytic cond. $691.823$
Root an. cond. $26.3025$
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  + 3-s − 5-s + 4·7-s + 9-s − 2·11-s − 6·13-s − 15-s − 2·17-s + 4·21-s − 6·23-s + 25-s + 27-s − 8·29-s − 8·31-s − 2·33-s − 4·35-s − 10·37-s − 6·39-s + 4·41-s − 4·43-s − 45-s − 6·47-s + 9·49-s − 2·51-s − 12·53-s + 2·55-s − 8·59-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s − 0.603·11-s − 1.66·13-s − 0.258·15-s − 0.485·17-s + 0.872·21-s − 1.25·23-s + 1/5·25-s + 0.192·27-s − 1.48·29-s − 1.43·31-s − 0.348·33-s − 0.676·35-s − 1.64·37-s − 0.960·39-s + 0.624·41-s − 0.609·43-s − 0.149·45-s − 0.875·47-s + 9/7·49-s − 0.280·51-s − 1.64·53-s + 0.269·55-s − 1.04·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 86640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(86640\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(691.823\)
Root analytic conductor: \(26.3025\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 86640,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7948190705\)
\(L(\frac12)\) \(\approx\) \(0.7948190705\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 + T \)
19 \( 1 \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
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

−14.17633893074109, −13.42268884063639, −12.89945636665329, −12.43320032436218, −11.93663631588431, −11.43419898701353, −10.95176027461753, −10.51306874296345, −9.857993817595649, −9.396248373391720, −8.816548112960635, −8.246985784344063, −7.809452790645293, −7.450618744900147, −7.103467080098868, −6.208604803862543, −5.418489492464161, −4.973228471701792, −4.634277498972228, −3.899789355110170, −3.369522045027907, −2.530698891478628, −1.828544029035952, −1.749136156060141, −0.2505947682201484, 0.2505947682201484, 1.749136156060141, 1.828544029035952, 2.530698891478628, 3.369522045027907, 3.899789355110170, 4.634277498972228, 4.973228471701792, 5.418489492464161, 6.208604803862543, 7.103467080098868, 7.450618744900147, 7.809452790645293, 8.246985784344063, 8.816548112960635, 9.396248373391720, 9.857993817595649, 10.51306874296345, 10.95176027461753, 11.43419898701353, 11.93663631588431, 12.43320032436218, 12.89945636665329, 13.42268884063639, 14.17633893074109

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