Properties

Label 2-133e2-1.1-c1-0-2
Degree $2$
Conductor $17689$
Sign $-1$
Analytic cond. $141.247$
Root an. cond. $11.8847$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·4-s − 3·5-s + 9-s + 3·11-s + 4·12-s − 4·13-s + 6·15-s + 4·16-s + 3·17-s + 6·20-s + 4·25-s + 4·27-s − 6·29-s − 4·31-s − 6·33-s − 2·36-s − 2·37-s + 8·39-s − 6·41-s − 43-s − 6·44-s − 3·45-s + 3·47-s − 8·48-s − 6·51-s + 8·52-s + ⋯
L(s)  = 1  − 1.15·3-s − 4-s − 1.34·5-s + 1/3·9-s + 0.904·11-s + 1.15·12-s − 1.10·13-s + 1.54·15-s + 16-s + 0.727·17-s + 1.34·20-s + 4/5·25-s + 0.769·27-s − 1.11·29-s − 0.718·31-s − 1.04·33-s − 1/3·36-s − 0.328·37-s + 1.28·39-s − 0.937·41-s − 0.152·43-s − 0.904·44-s − 0.447·45-s + 0.437·47-s − 1.15·48-s − 0.840·51-s + 1.10·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17689 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17689 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17689\)    =    \(7^{2} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(141.247\)
Root analytic conductor: \(11.8847\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 17689,\ (\ :1/2),\ -1)\)

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)$
bad7 \( 1 \)
19 \( 1 \)
good2 \( 1 + p T^{2} \)
3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−16.24062917513488, −15.58788124547474, −14.97625082444832, −14.40949841703842, −14.15230409835509, −13.14532597696508, −12.51359672009654, −12.25815090454689, −11.64927693836599, −11.32739203066337, −10.58331962908708, −9.957876800900652, −9.358863964568489, −8.757498864886458, −8.082079215054112, −7.468552785540221, −7.015775804758812, −6.103219812061688, −5.528913749029299, −4.833805940802903, −4.465478507359845, −3.637043445132515, −3.209213954492848, −1.704338888578465, −0.6443735371877729, 0, 0.6443735371877729, 1.704338888578465, 3.209213954492848, 3.637043445132515, 4.465478507359845, 4.833805940802903, 5.528913749029299, 6.103219812061688, 7.015775804758812, 7.468552785540221, 8.082079215054112, 8.757498864886458, 9.358863964568489, 9.957876800900652, 10.58331962908708, 11.32739203066337, 11.64927693836599, 12.25815090454689, 12.51359672009654, 13.14532597696508, 14.15230409835509, 14.40949841703842, 14.97625082444832, 15.58788124547474, 16.24062917513488

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