Properties

Label 2-98-1.1-c1-0-1
Degree $2$
Conductor $98$
Sign $1$
Analytic cond. $0.782533$
Root an. cond. $0.884609$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 1.41·3-s + 4-s + 2.82·5-s − 1.41·6-s + 8-s − 0.999·9-s + 2.82·10-s − 2·11-s − 1.41·12-s − 4.00·15-s + 16-s − 1.41·17-s − 0.999·18-s − 7.07·19-s + 2.82·20-s − 2·22-s − 4·23-s − 1.41·24-s + 3.00·25-s + 5.65·27-s + 2·29-s − 4.00·30-s + 8.48·31-s + 32-s + 2.82·33-s − 1.41·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.816·3-s + 0.5·4-s + 1.26·5-s − 0.577·6-s + 0.353·8-s − 0.333·9-s + 0.894·10-s − 0.603·11-s − 0.408·12-s − 1.03·15-s + 0.250·16-s − 0.342·17-s − 0.235·18-s − 1.62·19-s + 0.632·20-s − 0.426·22-s − 0.834·23-s − 0.288·24-s + 0.600·25-s + 1.08·27-s + 0.371·29-s − 0.730·30-s + 1.52·31-s + 0.176·32-s + 0.492·33-s − 0.242·34-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.782533\)
Root analytic conductor: \(0.884609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{98} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.278450789\)
\(L(\frac12)\) \(\approx\) \(1.278450789\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
7 \( 1 \)
good3 \( 1 + 1.41T + 3T^{2} \)
5 \( 1 - 2.82T + 5T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 1.41T + 17T^{2} \)
19 \( 1 + 7.07T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 - 8.48T + 31T^{2} \)
37 \( 1 - 10T + 37T^{2} \)
41 \( 1 + 9.89T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 + 1.41T + 59T^{2} \)
61 \( 1 - 2.82T + 61T^{2} \)
67 \( 1 - 12T + 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + 1.41T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 9.89T + 83T^{2} \)
89 \( 1 + 7.07T + 89T^{2} \)
97 \( 1 - 9.89T + 97T^{2} \)
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

−13.75897892764682158104944478298, −12.99181020642601878454218038210, −11.91815093620013056118481872743, −10.79739703594736369657619807538, −9.981854834006967455041537620725, −8.371218444153554891246935980326, −6.48255113005534387242997547576, −5.86451827160416506222883333068, −4.66415541315340900968485969565, −2.41717302449154006608420706146, 2.41717302449154006608420706146, 4.66415541315340900968485969565, 5.86451827160416506222883333068, 6.48255113005534387242997547576, 8.371218444153554891246935980326, 9.981854834006967455041537620725, 10.79739703594736369657619807538, 11.91815093620013056118481872743, 12.99181020642601878454218038210, 13.75897892764682158104944478298

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