Properties

Label 2-9196-1.1-c1-0-78
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  − 3.31·3-s + 1.60·5-s + 4.11·7-s + 8.00·9-s + 0.192·13-s − 5.33·15-s + 7.62·17-s + 19-s − 13.6·21-s + 6.73·23-s − 2.41·25-s − 16.5·27-s − 6.80·29-s + 7.33·31-s + 6.61·35-s + 0.734·37-s − 0.637·39-s − 0.976·41-s + 5.55·43-s + 12.8·45-s + 8.83·47-s + 9.90·49-s − 25.2·51-s + 11.8·53-s − 3.31·57-s + 3.30·59-s − 9.41·61-s + ⋯
L(s)  = 1  − 1.91·3-s + 0.719·5-s + 1.55·7-s + 2.66·9-s + 0.0533·13-s − 1.37·15-s + 1.84·17-s + 0.229·19-s − 2.97·21-s + 1.40·23-s − 0.482·25-s − 3.19·27-s − 1.26·29-s + 1.31·31-s + 1.11·35-s + 0.120·37-s − 0.102·39-s − 0.152·41-s + 0.847·43-s + 1.91·45-s + 1.28·47-s + 1.41·49-s − 3.54·51-s + 1.62·53-s − 0.439·57-s + 0.429·59-s − 1.20·61-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

\(L(1)\) \(\approx\) \(1.976590320\)
\(L(\frac12)\) \(\approx\) \(1.976590320\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.31T + 3T^{2} \)
5 \( 1 - 1.60T + 5T^{2} \)
7 \( 1 - 4.11T + 7T^{2} \)
13 \( 1 - 0.192T + 13T^{2} \)
17 \( 1 - 7.62T + 17T^{2} \)
23 \( 1 - 6.73T + 23T^{2} \)
29 \( 1 + 6.80T + 29T^{2} \)
31 \( 1 - 7.33T + 31T^{2} \)
37 \( 1 - 0.734T + 37T^{2} \)
41 \( 1 + 0.976T + 41T^{2} \)
43 \( 1 - 5.55T + 43T^{2} \)
47 \( 1 - 8.83T + 47T^{2} \)
53 \( 1 - 11.8T + 53T^{2} \)
59 \( 1 - 3.30T + 59T^{2} \)
61 \( 1 + 9.41T + 61T^{2} \)
67 \( 1 - 12.5T + 67T^{2} \)
71 \( 1 - 4.12T + 71T^{2} \)
73 \( 1 - 6.15T + 73T^{2} \)
79 \( 1 - 2.98T + 79T^{2} \)
83 \( 1 + 6.33T + 83T^{2} \)
89 \( 1 + 13.5T + 89T^{2} \)
97 \( 1 + 4.33T + 97T^{2} \)
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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

−7.49794203634688313914661681570, −7.03714230586755954030667106200, −6.06524947537870371967134779236, −5.50967985498116893358501199457, −5.27479399569336077044075363375, −4.54739056431533960628620540023, −3.74880922899040425221968544352, −2.31011876214535916601158543362, −1.29756604352299163656107365630, −0.917487193278197447093505296668, 0.917487193278197447093505296668, 1.29756604352299163656107365630, 2.31011876214535916601158543362, 3.74880922899040425221968544352, 4.54739056431533960628620540023, 5.27479399569336077044075363375, 5.50967985498116893358501199457, 6.06524947537870371967134779236, 7.03714230586755954030667106200, 7.49794203634688313914661681570

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