Properties

Label 2-98315-1.1-c1-0-5
Degree $2$
Conductor $98315$
Sign $-1$
Analytic cond. $785.049$
Root an. cond. $28.0187$
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  − 3-s − 2·4-s + 5-s + 7-s − 2·9-s − 3·11-s + 2·12-s + 5·13-s − 15-s + 4·16-s + 3·17-s − 2·19-s − 2·20-s − 21-s + 6·23-s + 25-s + 5·27-s − 2·28-s + 3·29-s + 4·31-s + 3·33-s + 35-s + 4·36-s + 2·37-s − 5·39-s + 12·41-s − 10·43-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s + 0.447·5-s + 0.377·7-s − 2/3·9-s − 0.904·11-s + 0.577·12-s + 1.38·13-s − 0.258·15-s + 16-s + 0.727·17-s − 0.458·19-s − 0.447·20-s − 0.218·21-s + 1.25·23-s + 1/5·25-s + 0.962·27-s − 0.377·28-s + 0.557·29-s + 0.718·31-s + 0.522·33-s + 0.169·35-s + 2/3·36-s + 0.328·37-s − 0.800·39-s + 1.87·41-s − 1.52·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98315\)    =    \(5 \cdot 7 \cdot 53^{2}\)
Sign: $-1$
Analytic conductor: \(785.049\)
Root analytic conductor: \(28.0187\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{98315} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 98315,\ (\ :1/2),\ -1)\)

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)$
bad5 \( 1 - T \)
7 \( 1 - T \)
53 \( 1 \)
good2 \( 1 + p T^{2} \)
3 \( 1 + T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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

−13.85346950558918, −13.67570533130117, −12.95983655125841, −12.71537416840034, −12.13288847272401, −11.48463019222317, −10.95268501360649, −10.67994076881908, −10.13667863992613, −9.586195017775741, −8.939619175086594, −8.547211800953863, −8.207760904469649, −7.613517284064230, −6.881631340689051, −6.151652884239162, −5.767942911398682, −5.423030522016237, −4.751145341690580, −4.407973423891949, −3.553564399221230, −2.987518718978241, −2.421565215462578, −1.226726781240785, −0.9569845106214991, 0, 0.9569845106214991, 1.226726781240785, 2.421565215462578, 2.987518718978241, 3.553564399221230, 4.407973423891949, 4.751145341690580, 5.423030522016237, 5.767942911398682, 6.151652884239162, 6.881631340689051, 7.613517284064230, 8.207760904469649, 8.547211800953863, 8.939619175086594, 9.586195017775741, 10.13667863992613, 10.67994076881908, 10.95268501360649, 11.48463019222317, 12.13288847272401, 12.71537416840034, 12.95983655125841, 13.67570533130117, 13.85346950558918

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