Properties

Label 2-136e2-1.1-c1-0-13
Degree $2$
Conductor $18496$
Sign $-1$
Analytic cond. $147.691$
Root an. cond. $12.1528$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·7-s + 9-s + 2·11-s + 2·13-s + 4·19-s + 4·21-s + 6·23-s − 5·25-s + 4·27-s + 8·29-s − 6·31-s − 4·33-s − 8·37-s − 4·39-s − 12·43-s − 8·47-s − 3·49-s + 6·53-s − 8·57-s + 4·59-s + 8·61-s − 2·63-s − 4·67-s − 12·69-s − 6·71-s + 8·73-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.755·7-s + 1/3·9-s + 0.603·11-s + 0.554·13-s + 0.917·19-s + 0.872·21-s + 1.25·23-s − 25-s + 0.769·27-s + 1.48·29-s − 1.07·31-s − 0.696·33-s − 1.31·37-s − 0.640·39-s − 1.82·43-s − 1.16·47-s − 3/7·49-s + 0.824·53-s − 1.05·57-s + 0.520·59-s + 1.02·61-s − 0.251·63-s − 0.488·67-s − 1.44·69-s − 0.712·71-s + 0.936·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(18496\)    =    \(2^{6} \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(147.691\)
Root analytic conductor: \(12.1528\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 18496,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−16.10402210182327, −15.75315179424372, −15.01663583952794, −14.39100593626232, −13.75419844063102, −13.23264105637803, −12.68385745181409, −12.02000745622450, −11.60957542557561, −11.23110055607009, −10.45820494197507, −10.01424178665542, −9.410003065597100, −8.708053335595257, −8.208109640238114, −7.139847164709316, −6.751412365189894, −6.314533654586069, −5.476119755821194, −5.202680335037429, −4.329377712937530, −3.439297216367321, −3.046257440295446, −1.746315123648533, −0.9403820759880154, 0, 0.9403820759880154, 1.746315123648533, 3.046257440295446, 3.439297216367321, 4.329377712937530, 5.202680335037429, 5.476119755821194, 6.314533654586069, 6.751412365189894, 7.139847164709316, 8.208109640238114, 8.708053335595257, 9.410003065597100, 10.01424178665542, 10.45820494197507, 11.23110055607009, 11.60957542557561, 12.02000745622450, 12.68385745181409, 13.23264105637803, 13.75419844063102, 14.39100593626232, 15.01663583952794, 15.75315179424372, 16.10402210182327

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