Properties

Label 2-9196-1.1-c1-0-54
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.82·3-s − 0.343·5-s − 4.77·7-s + 4.98·9-s − 1.86·13-s − 0.970·15-s + 5.69·17-s + 19-s − 13.4·21-s − 1.06·23-s − 4.88·25-s + 5.60·27-s + 9.31·29-s + 2.97·31-s + 1.64·35-s − 3.54·37-s − 5.26·39-s − 2.82·41-s − 8.09·43-s − 1.71·45-s + 3.00·47-s + 15.7·49-s + 16.0·51-s − 4.33·53-s + 2.82·57-s + 14.6·59-s − 1.03·61-s + ⋯
L(s)  = 1  + 1.63·3-s − 0.153·5-s − 1.80·7-s + 1.66·9-s − 0.516·13-s − 0.250·15-s + 1.38·17-s + 0.229·19-s − 2.94·21-s − 0.221·23-s − 0.976·25-s + 1.07·27-s + 1.72·29-s + 0.533·31-s + 0.277·35-s − 0.583·37-s − 0.843·39-s − 0.440·41-s − 1.23·43-s − 0.255·45-s + 0.438·47-s + 2.25·49-s + 2.25·51-s − 0.595·53-s + 0.374·57-s + 1.90·59-s − 0.132·61-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

\(L(1)\) \(\approx\) \(2.914419115\)
\(L(\frac12)\) \(\approx\) \(2.914419115\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.82T + 3T^{2} \)
5 \( 1 + 0.343T + 5T^{2} \)
7 \( 1 + 4.77T + 7T^{2} \)
13 \( 1 + 1.86T + 13T^{2} \)
17 \( 1 - 5.69T + 17T^{2} \)
23 \( 1 + 1.06T + 23T^{2} \)
29 \( 1 - 9.31T + 29T^{2} \)
31 \( 1 - 2.97T + 31T^{2} \)
37 \( 1 + 3.54T + 37T^{2} \)
41 \( 1 + 2.82T + 41T^{2} \)
43 \( 1 + 8.09T + 43T^{2} \)
47 \( 1 - 3.00T + 47T^{2} \)
53 \( 1 + 4.33T + 53T^{2} \)
59 \( 1 - 14.6T + 59T^{2} \)
61 \( 1 + 1.03T + 61T^{2} \)
67 \( 1 - 7.93T + 67T^{2} \)
71 \( 1 + 11.4T + 71T^{2} \)
73 \( 1 - 16.3T + 73T^{2} \)
79 \( 1 - 13.3T + 79T^{2} \)
83 \( 1 - 3.40T + 83T^{2} \)
89 \( 1 - 0.0562T + 89T^{2} \)
97 \( 1 - 11.4T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.893516689522797403537203472703, −7.07670270658053879338981755231, −6.60754421929732088333064401446, −5.76308736648524801566092890603, −4.80545765269407511065824673558, −3.74301728588056231539003333160, −3.41002590540326031575575348435, −2.81683814901502365136588976498, −2.05353532433346521566657175669, −0.73530211297840959185255807652, 0.73530211297840959185255807652, 2.05353532433346521566657175669, 2.81683814901502365136588976498, 3.41002590540326031575575348435, 3.74301728588056231539003333160, 4.80545765269407511065824673558, 5.76308736648524801566092890603, 6.60754421929732088333064401446, 7.07670270658053879338981755231, 7.893516689522797403537203472703

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