Properties

Label 2-9196-1.1-c1-0-14
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  − 0.566·3-s − 3.92·5-s + 2.44·7-s − 2.67·9-s − 6.76·13-s + 2.22·15-s + 0.175·17-s + 19-s − 1.38·21-s + 8.30·23-s + 10.4·25-s + 3.21·27-s + 0.843·29-s − 0.224·31-s − 9.60·35-s − 3.49·37-s + 3.83·39-s − 5.09·41-s − 9.48·43-s + 10.5·45-s − 9.68·47-s − 1.01·49-s − 0.0996·51-s − 4.72·53-s − 0.566·57-s − 7.88·59-s − 14.9·61-s + ⋯
L(s)  = 1  − 0.326·3-s − 1.75·5-s + 0.924·7-s − 0.893·9-s − 1.87·13-s + 0.574·15-s + 0.0426·17-s + 0.229·19-s − 0.302·21-s + 1.73·23-s + 2.08·25-s + 0.618·27-s + 0.156·29-s − 0.0403·31-s − 1.62·35-s − 0.574·37-s + 0.613·39-s − 0.795·41-s − 1.44·43-s + 1.56·45-s − 1.41·47-s − 0.145·49-s − 0.0139·51-s − 0.649·53-s − 0.0750·57-s − 1.02·59-s − 1.91·61-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

\(L(1)\) \(\approx\) \(0.4693097836\)
\(L(\frac12)\) \(\approx\) \(0.4693097836\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.566T + 3T^{2} \)
5 \( 1 + 3.92T + 5T^{2} \)
7 \( 1 - 2.44T + 7T^{2} \)
13 \( 1 + 6.76T + 13T^{2} \)
17 \( 1 - 0.175T + 17T^{2} \)
23 \( 1 - 8.30T + 23T^{2} \)
29 \( 1 - 0.843T + 29T^{2} \)
31 \( 1 + 0.224T + 31T^{2} \)
37 \( 1 + 3.49T + 37T^{2} \)
41 \( 1 + 5.09T + 41T^{2} \)
43 \( 1 + 9.48T + 43T^{2} \)
47 \( 1 + 9.68T + 47T^{2} \)
53 \( 1 + 4.72T + 53T^{2} \)
59 \( 1 + 7.88T + 59T^{2} \)
61 \( 1 + 14.9T + 61T^{2} \)
67 \( 1 - 2.09T + 67T^{2} \)
71 \( 1 + 7.53T + 71T^{2} \)
73 \( 1 - 1.36T + 73T^{2} \)
79 \( 1 - 1.04T + 79T^{2} \)
83 \( 1 + 2.37T + 83T^{2} \)
89 \( 1 - 3.38T + 89T^{2} \)
97 \( 1 - 19.0T + 97T^{2} \)
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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

−7.81552907798446794060564274141, −7.12164443088856811277274862859, −6.61431983190170516431360929211, −5.31685557581638079722315403109, −4.85887755522816424733105271139, −4.52819969639310580602240517054, −3.24361335283491094268522599549, −2.96492202204678729056406587700, −1.61679478065239104608485466535, −0.33265589163497194137293581398, 0.33265589163497194137293581398, 1.61679478065239104608485466535, 2.96492202204678729056406587700, 3.24361335283491094268522599549, 4.52819969639310580602240517054, 4.85887755522816424733105271139, 5.31685557581638079722315403109, 6.61431983190170516431360929211, 7.12164443088856811277274862859, 7.81552907798446794060564274141

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