Properties

Label 2-98-1.1-c13-0-14
Degree $2$
Conductor $98$
Sign $1$
Analytic cond. $105.086$
Root an. cond. $10.2511$
Motivic weight $13$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 64·2-s − 1.62e3·3-s + 4.09e3·4-s + 3.64e4·5-s + 1.04e5·6-s − 2.62e5·8-s + 1.04e6·9-s − 2.32e6·10-s + 2.60e6·11-s − 6.66e6·12-s + 1.26e7·13-s − 5.91e7·15-s + 1.67e7·16-s + 1.30e8·17-s − 6.71e7·18-s + 2.49e8·19-s + 1.49e8·20-s − 1.66e8·22-s + 4.89e8·23-s + 4.26e8·24-s + 1.04e8·25-s − 8.07e8·26-s + 8.85e8·27-s − 1.12e8·29-s + 3.78e9·30-s + 9.10e9·31-s − 1.07e9·32-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.28·3-s + 1/2·4-s + 1.04·5-s + 0.910·6-s − 0.353·8-s + 0.658·9-s − 0.736·10-s + 0.443·11-s − 0.643·12-s + 0.725·13-s − 1.34·15-s + 1/4·16-s + 1.31·17-s − 0.465·18-s + 1.21·19-s + 0.520·20-s − 0.313·22-s + 0.688·23-s + 0.455·24-s + 0.0854·25-s − 0.512·26-s + 0.440·27-s − 0.0350·29-s + 0.948·30-s + 1.84·31-s − 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(105.086\)
Root analytic conductor: \(10.2511\)
Motivic weight: \(13\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :13/2),\ 1)\)

Particular Values

\(L(7)\) \(\approx\) \(1.498652583\)
\(L(\frac12)\) \(\approx\) \(1.498652583\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p^{6} T \)
7 \( 1 \)
good3 \( 1 + 542 p T + p^{13} T^{2} \)
5 \( 1 - 1456 p^{2} T + p^{13} T^{2} \)
11 \( 1 - 2605288 T + p^{13} T^{2} \)
13 \( 1 - 12624468 T + p^{13} T^{2} \)
17 \( 1 - 130752362 T + p^{13} T^{2} \)
19 \( 1 - 249436042 T + p^{13} T^{2} \)
23 \( 1 - 489054160 T + p^{13} T^{2} \)
29 \( 1 + 112115926 T + p^{13} T^{2} \)
31 \( 1 - 9103068684 T + p^{13} T^{2} \)
37 \( 1 - 18308169938 T + p^{13} T^{2} \)
41 \( 1 + 13082373606 T + p^{13} T^{2} \)
43 \( 1 + 67123460032 T + p^{13} T^{2} \)
47 \( 1 + 105239980284 T + p^{13} T^{2} \)
53 \( 1 + 25221720042 T + p^{13} T^{2} \)
59 \( 1 - 276774602098 T + p^{13} T^{2} \)
61 \( 1 + 759388645560 T + p^{13} T^{2} \)
67 \( 1 - 1039664575708 T + p^{13} T^{2} \)
71 \( 1 - 1817086195456 T + p^{13} T^{2} \)
73 \( 1 + 400342248850 T + p^{13} T^{2} \)
79 \( 1 + 3597798513336 T + p^{13} T^{2} \)
83 \( 1 - 1309030493954 T + p^{13} T^{2} \)
89 \( 1 + 1653288354570 T + p^{13} T^{2} \)
97 \( 1 - 12736909073690 T + p^{13} T^{2} \)
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

−11.34791097515217237850959153464, −10.17596589804361857796091742612, −9.577445257969757317191275861738, −8.153480686336163779761674558948, −6.69011341472140124502068864115, −5.95091611363439847084217395159, −5.05578275980564987059119160804, −3.11957936971419670478347691379, −1.44678908252174879750618255924, −0.78840785813436426606307655351, 0.78840785813436426606307655351, 1.44678908252174879750618255924, 3.11957936971419670478347691379, 5.05578275980564987059119160804, 5.95091611363439847084217395159, 6.69011341472140124502068864115, 8.153480686336163779761674558948, 9.577445257969757317191275861738, 10.17596589804361857796091742612, 11.34791097515217237850959153464

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