Properties

Label 2-9196-1.1-c1-0-13
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  − 2.05·3-s + 0.680·5-s − 1.59·7-s + 1.23·9-s + 0.546·13-s − 1.39·15-s − 6.61·17-s + 19-s + 3.27·21-s − 6.44·23-s − 4.53·25-s + 3.62·27-s − 7.32·29-s + 3.39·31-s − 1.08·35-s + 8.83·37-s − 1.12·39-s − 3.18·41-s − 10.4·43-s + 0.841·45-s − 7.25·47-s − 4.46·49-s + 13.6·51-s + 7.47·53-s − 2.05·57-s + 11.1·59-s − 8.75·61-s + ⋯
L(s)  = 1  − 1.18·3-s + 0.304·5-s − 0.601·7-s + 0.412·9-s + 0.151·13-s − 0.361·15-s − 1.60·17-s + 0.229·19-s + 0.714·21-s − 1.34·23-s − 0.907·25-s + 0.698·27-s − 1.35·29-s + 0.610·31-s − 0.182·35-s + 1.45·37-s − 0.180·39-s − 0.497·41-s − 1.58·43-s + 0.125·45-s − 1.05·47-s − 0.638·49-s + 1.90·51-s + 1.02·53-s − 0.272·57-s + 1.44·59-s − 1.12·61-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

\(L(1)\) \(\approx\) \(0.4376077186\)
\(L(\frac12)\) \(\approx\) \(0.4376077186\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.05T + 3T^{2} \)
5 \( 1 - 0.680T + 5T^{2} \)
7 \( 1 + 1.59T + 7T^{2} \)
13 \( 1 - 0.546T + 13T^{2} \)
17 \( 1 + 6.61T + 17T^{2} \)
23 \( 1 + 6.44T + 23T^{2} \)
29 \( 1 + 7.32T + 29T^{2} \)
31 \( 1 - 3.39T + 31T^{2} \)
37 \( 1 - 8.83T + 37T^{2} \)
41 \( 1 + 3.18T + 41T^{2} \)
43 \( 1 + 10.4T + 43T^{2} \)
47 \( 1 + 7.25T + 47T^{2} \)
53 \( 1 - 7.47T + 53T^{2} \)
59 \( 1 - 11.1T + 59T^{2} \)
61 \( 1 + 8.75T + 61T^{2} \)
67 \( 1 - 11.1T + 67T^{2} \)
71 \( 1 + 7.88T + 71T^{2} \)
73 \( 1 + 3.00T + 73T^{2} \)
79 \( 1 + 8.73T + 79T^{2} \)
83 \( 1 - 7.76T + 83T^{2} \)
89 \( 1 - 15.7T + 89T^{2} \)
97 \( 1 - 9.99T + 97T^{2} \)
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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

−7.64444584593781078325211755554, −6.64460636356149404980453306971, −6.39475348920415827658616963016, −5.76913313105449664737756715485, −5.09648388404424107382947493376, −4.31827773513615233529637753670, −3.59737750447278896843902853793, −2.50075095351376076360500111574, −1.68204206098905578555238127514, −0.32832648709508235527474515308, 0.32832648709508235527474515308, 1.68204206098905578555238127514, 2.50075095351376076360500111574, 3.59737750447278896843902853793, 4.31827773513615233529637753670, 5.09648388404424107382947493376, 5.76913313105449664737756715485, 6.39475348920415827658616963016, 6.64460636356149404980453306971, 7.64444584593781078325211755554

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