Properties

Label 2-9196-1.1-c1-0-137
Degree $2$
Conductor $9196$
Sign $1$
Analytic cond. $73.4304$
Root an. cond. $8.56915$
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  + 3.27·3-s + 3.53·5-s + 2.34·7-s + 7.75·9-s + 5.44·13-s + 11.5·15-s − 3.37·17-s + 19-s + 7.68·21-s − 5.78·23-s + 7.49·25-s + 15.5·27-s − 9.19·29-s − 9.59·31-s + 8.28·35-s − 6.58·37-s + 17.8·39-s − 2.37·41-s − 1.18·43-s + 27.4·45-s + 1.69·47-s − 1.50·49-s − 11.0·51-s + 2.51·53-s + 3.27·57-s − 4.83·59-s + 7.62·61-s + ⋯
L(s)  = 1  + 1.89·3-s + 1.58·5-s + 0.886·7-s + 2.58·9-s + 1.50·13-s + 2.99·15-s − 0.818·17-s + 0.229·19-s + 1.67·21-s − 1.20·23-s + 1.49·25-s + 2.99·27-s − 1.70·29-s − 1.72·31-s + 1.40·35-s − 1.08·37-s + 2.85·39-s − 0.370·41-s − 0.179·43-s + 4.08·45-s + 0.246·47-s − 0.214·49-s − 1.55·51-s + 0.344·53-s + 0.434·57-s − 0.630·59-s + 0.976·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9196\)    =    \(2^{2} \cdot 11^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(73.4304\)
Root analytic conductor: \(8.56915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9196,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.270318420\)
\(L(\frac12)\) \(\approx\) \(7.270318420\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
19 \( 1 - T \)
good3 \( 1 - 3.27T + 3T^{2} \)
5 \( 1 - 3.53T + 5T^{2} \)
7 \( 1 - 2.34T + 7T^{2} \)
13 \( 1 - 5.44T + 13T^{2} \)
17 \( 1 + 3.37T + 17T^{2} \)
23 \( 1 + 5.78T + 23T^{2} \)
29 \( 1 + 9.19T + 29T^{2} \)
31 \( 1 + 9.59T + 31T^{2} \)
37 \( 1 + 6.58T + 37T^{2} \)
41 \( 1 + 2.37T + 41T^{2} \)
43 \( 1 + 1.18T + 43T^{2} \)
47 \( 1 - 1.69T + 47T^{2} \)
53 \( 1 - 2.51T + 53T^{2} \)
59 \( 1 + 4.83T + 59T^{2} \)
61 \( 1 - 7.62T + 61T^{2} \)
67 \( 1 - 2.22T + 67T^{2} \)
71 \( 1 - 11.0T + 71T^{2} \)
73 \( 1 - 14.9T + 73T^{2} \)
79 \( 1 - 5.18T + 79T^{2} \)
83 \( 1 - 4.24T + 83T^{2} \)
89 \( 1 - 6.09T + 89T^{2} \)
97 \( 1 + 6.16T + 97T^{2} \)
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

−7.962990179369471332200617148159, −7.15010645978953954874499529441, −6.46296677358641858917714130363, −5.63714473990472762406980111772, −4.95471884028393544043679425165, −3.80012742334194740895557194396, −3.58918191433433462474299929352, −2.27060655434224588389586979573, −1.95295852371940405162471581926, −1.40910753242949966180163605166, 1.40910753242949966180163605166, 1.95295852371940405162471581926, 2.27060655434224588389586979573, 3.58918191433433462474299929352, 3.80012742334194740895557194396, 4.95471884028393544043679425165, 5.63714473990472762406980111772, 6.46296677358641858917714130363, 7.15010645978953954874499529441, 7.962990179369471332200617148159

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