Properties

Label 2-97104-1.1-c1-0-76
Degree $2$
Conductor $97104$
Sign $1$
Analytic cond. $775.379$
Root an. cond. $27.8456$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2·5-s − 7-s + 9-s − 6·13-s − 2·15-s + 21-s − 8·23-s − 25-s − 27-s + 6·29-s − 8·31-s − 2·35-s − 10·37-s + 6·39-s + 6·41-s − 12·43-s + 2·45-s + 49-s − 10·53-s + 8·59-s − 6·61-s − 63-s − 12·65-s − 12·67-s + 8·69-s + 6·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s − 1.66·13-s − 0.516·15-s + 0.218·21-s − 1.66·23-s − 1/5·25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.338·35-s − 1.64·37-s + 0.960·39-s + 0.937·41-s − 1.82·43-s + 0.298·45-s + 1/7·49-s − 1.37·53-s + 1.04·59-s − 0.768·61-s − 0.125·63-s − 1.48·65-s − 1.46·67-s + 0.963·69-s + 0.702·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(97104\)    =    \(2^{4} \cdot 3 \cdot 7 \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(775.379\)
Root analytic conductor: \(27.8456\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{97104} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 97104,\ (\ :1/2),\ 1)\)

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 \( 1 + T \)
7 \( 1 + T \)
17 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−14.23092230741435, −13.88315838299135, −13.23351556180195, −12.71271542402531, −12.35859089067632, −11.85958574955932, −11.49818565564572, −10.62683334261201, −10.27030699635793, −9.876722540292914, −9.547065125489968, −8.930675804514058, −8.275616107423727, −7.640648714094864, −7.166869409344628, −6.612835362600140, −6.142730592823840, −5.561893576772493, −5.174725476665906, −4.583855134763415, −3.957557344993546, −3.212869239345866, −2.524108510112134, −1.913789970967884, −1.432185671906986, 0, 0, 1.432185671906986, 1.913789970967884, 2.524108510112134, 3.212869239345866, 3.957557344993546, 4.583855134763415, 5.174725476665906, 5.561893576772493, 6.142730592823840, 6.612835362600140, 7.166869409344628, 7.640648714094864, 8.275616107423727, 8.930675804514058, 9.547065125489968, 9.876722540292914, 10.27030699635793, 10.62683334261201, 11.49818565564572, 11.85958574955932, 12.35859089067632, 12.71271542402531, 13.23351556180195, 13.88315838299135, 14.23092230741435

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