Properties

Label 2-14490-1.1-c1-0-47
Degree $2$
Conductor $14490$
Sign $-1$
Analytic cond. $115.703$
Root an. cond. $10.7565$
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-s + 4-s − 5-s + 7-s + 8-s − 10-s + 2·13-s + 14-s + 16-s − 4·19-s − 20-s + 23-s + 25-s + 2·26-s + 28-s − 4·31-s + 32-s − 35-s + 2·37-s − 4·38-s − 40-s − 12·41-s − 4·43-s + 46-s − 12·47-s + 49-s + 50-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s + 0.353·8-s − 0.316·10-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.917·19-s − 0.223·20-s + 0.208·23-s + 1/5·25-s + 0.392·26-s + 0.188·28-s − 0.718·31-s + 0.176·32-s − 0.169·35-s + 0.328·37-s − 0.648·38-s − 0.158·40-s − 1.87·41-s − 0.609·43-s + 0.147·46-s − 1.75·47-s + 1/7·49-s + 0.141·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14490\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(115.703\)
Root analytic conductor: \(10.7565\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 14490,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
3 \( 1 \)
5 \( 1 + T \)
7 \( 1 - T \)
23 \( 1 - T \)
good11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−16.42259932777921, −15.67217678442771, −15.12811103157805, −14.81473872153861, −14.19459939352097, −13.49826160112518, −13.11366362517170, −12.47156779905833, −11.93644923389275, −11.26616360426692, −10.97434473159672, −10.26763730726516, −9.587989066664940, −8.724030369114870, −8.235157376957419, −7.715596135306754, −6.705933503623747, −6.593642401008249, −5.563551150192077, −5.032376704866590, −4.345080926984466, −3.681912217324908, −3.084730710240168, −2.085145004951509, −1.361686951557828, 0, 1.361686951557828, 2.085145004951509, 3.084730710240168, 3.681912217324908, 4.345080926984466, 5.032376704866590, 5.563551150192077, 6.593642401008249, 6.705933503623747, 7.715596135306754, 8.235157376957419, 8.724030369114870, 9.587989066664940, 10.26763730726516, 10.97434473159672, 11.26616360426692, 11.93644923389275, 12.47156779905833, 13.11366362517170, 13.49826160112518, 14.19459939352097, 14.81473872153861, 15.12811103157805, 15.67217678442771, 16.42259932777921

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