↑

Properties

Label 2-408e2-1.1-c1-0-138
Degree $2$
Conductor $166464$
Sign $-1$
Analytic cond. $1329.22$
Root an. cond. $36.4584$
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  + 3·5-s − 4·7-s + 3·11-s − 2·13-s + 8·19-s − 6·23-s + 4·25-s − 3·29-s − 7·31-s − 12·35-s + 8·37-s + 6·41-s − 4·43-s − 6·47-s + 9·49-s − 9·53-s + 9·55-s − 15·59-s + 14·61-s − 6·65-s + 2·67-s + 7·73-s − 12·77-s − 79-s − 12·83-s + 8·91-s + 24·95-s + ⋯
L(s)  = 1  + 1.34·5-s − 1.51·7-s + 0.904·11-s − 0.554·13-s + 1.83·19-s − 1.25·23-s + 4/5·25-s − 0.557·29-s − 1.25·31-s − 2.02·35-s + 1.31·37-s + 0.937·41-s − 0.609·43-s − 0.875·47-s + 9/7·49-s − 1.23·53-s + 1.21·55-s − 1.95·59-s + 1.79·61-s − 0.744·65-s + 0.244·67-s + 0.819·73-s − 1.36·77-s − 0.112·79-s − 1.31·83-s + 0.838·91-s + 2.46·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 166464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 166464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(166464\)    =    \(2^{6} \cdot 3^{2} \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(1329.22\)
Root analytic conductor: \(36.4584\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 166464,\ (\ :1/2),\ -1)\)

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 \( 1 \)
17 \( 1 \)
good5 \( 1 - 3 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 15 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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

−13.37376907021061, −13.00971898819743, −12.69275840884753, −12.15498778973762, −11.53577816390597, −11.24555341334174, −10.37658752558845, −9.944119905003256, −9.627077045292930, −9.334107179997487, −9.087465447624037, −8.131874623283640, −7.556737850434107, −7.130385139450764, −6.446715787666282, −6.192735642392759, −5.716460148409376, −5.271444943932472, −4.553929328095295, −3.808991321086613, −3.346951257280318, −2.852137516967059, −2.133229156154835, −1.628696695525455, −0.8631268199320170, 0, 0.8631268199320170, 1.628696695525455, 2.133229156154835, 2.852137516967059, 3.346951257280318, 3.808991321086613, 4.553929328095295, 5.271444943932472, 5.716460148409376, 6.192735642392759, 6.446715787666282, 7.130385139450764, 7.556737850434107, 8.131874623283640, 9.087465447624037, 9.334107179997487, 9.627077045292930, 9.944119905003256, 10.37658752558845, 11.24555341334174, 11.53577816390597, 12.15498778973762, 12.69275840884753, 13.00971898819743, 13.37376907021061

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