↑

Properties

Label 2-136e2-1.1-c1-0-11
Degree $2$
Conductor $18496$
Sign $-1$
Analytic cond. $147.691$
Root an. cond. $12.1528$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 4·7-s − 3·9-s + 2·13-s + 4·19-s − 4·23-s − 25-s + 6·29-s − 4·31-s + 8·35-s − 2·37-s + 6·41-s − 4·43-s + 6·45-s + 9·49-s − 6·53-s + 12·59-s − 10·61-s + 12·63-s − 4·65-s − 4·67-s + 4·71-s + 6·73-s − 12·79-s + 9·81-s + 4·83-s + 10·89-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.51·7-s − 9-s + 0.554·13-s + 0.917·19-s − 0.834·23-s − 1/5·25-s + 1.11·29-s − 0.718·31-s + 1.35·35-s − 0.328·37-s + 0.937·41-s − 0.609·43-s + 0.894·45-s + 9/7·49-s − 0.824·53-s + 1.56·59-s − 1.28·61-s + 1.51·63-s − 0.496·65-s − 0.488·67-s + 0.474·71-s + 0.702·73-s − 1.35·79-s + 81-s + 0.439·83-s + 1.05·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(18496\)    =    \(2^{6} \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(147.691\)
Root analytic conductor: \(12.1528\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 18496,\ (\ :1/2),\ -1)\)

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 \( 1 \)
17 \( 1 \)
good3 \( 1 + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
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 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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

−15.95463816799088, −15.73408956611726, −15.07250273522383, −14.26567764396787, −13.92227945607279, −13.29105106895295, −12.67814818571134, −12.14934624327076, −11.64952585348967, −11.19223196830903, −10.41926865187464, −9.892692630433800, −9.262729233857938, −8.720690683508882, −8.096734777306158, −7.524137854670346, −6.849346317886154, −6.130783524516034, −5.819051144767998, −4.911468363088498, −4.048404728552018, −3.364170043792341, −3.122832569468385, −2.129621386982329, −0.7894811583134163, 0, 0.7894811583134163, 2.129621386982329, 3.122832569468385, 3.364170043792341, 4.048404728552018, 4.911468363088498, 5.819051144767998, 6.130783524516034, 6.849346317886154, 7.524137854670346, 8.096734777306158, 8.720690683508882, 9.262729233857938, 9.892692630433800, 10.41926865187464, 11.19223196830903, 11.64952585348967, 12.14934624327076, 12.67814818571134, 13.29105106895295, 13.92227945607279, 14.26567764396787, 15.07250273522383, 15.73408956611726, 15.95463816799088

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