Properties

Label 2-229320-1.1-c1-0-116
Degree $2$
Conductor $229320$
Sign $-1$
Analytic cond. $1831.12$
Root an. cond. $42.7916$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 2·11-s − 13-s + 6·17-s + 6·23-s + 25-s − 8·31-s − 6·37-s − 2·41-s − 4·43-s − 8·53-s + 2·55-s − 8·59-s + 10·61-s − 65-s + 4·67-s − 6·71-s + 14·73-s − 8·79-s + 4·83-s + 6·85-s − 10·89-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.603·11-s − 0.277·13-s + 1.45·17-s + 1.25·23-s + 1/5·25-s − 1.43·31-s − 0.986·37-s − 0.312·41-s − 0.609·43-s − 1.09·53-s + 0.269·55-s − 1.04·59-s + 1.28·61-s − 0.124·65-s + 0.488·67-s − 0.712·71-s + 1.63·73-s − 0.900·79-s + 0.439·83-s + 0.650·85-s − 1.05·89-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(229320\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(1831.12\)
Root analytic conductor: \(42.7916\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 229320,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 \)
5 \( 1 - T \)
7 \( 1 \)
13 \( 1 + T \)
good11 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.06245957334554, −12.68514699905240, −12.32996267317919, −11.77999621366339, −11.30916767114170, −10.85735460263973, −10.30054304573571, −9.940801635513192, −9.294940807793181, −9.161730373203604, −8.492188219410427, −7.959816624005174, −7.468163709633942, −6.893554718913237, −6.624310223550041, −5.903391420694160, −5.350839942212068, −5.142450548086483, −4.453485607265319, −3.692301350790833, −3.344132848363989, −2.814132275259215, −1.987579848871117, −1.492358748458101, −0.9249928654932329, 0, 0.9249928654932329, 1.492358748458101, 1.987579848871117, 2.814132275259215, 3.344132848363989, 3.692301350790833, 4.453485607265319, 5.142450548086483, 5.350839942212068, 5.903391420694160, 6.624310223550041, 6.893554718913237, 7.468163709633942, 7.959816624005174, 8.492188219410427, 9.161730373203604, 9.294940807793181, 9.940801635513192, 10.30054304573571, 10.85735460263973, 11.30916767114170, 11.77999621366339, 12.32996267317919, 12.68514699905240, 13.06245957334554

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