Properties

Label 2-440e2-1.1-c1-0-73
Degree $2$
Conductor $193600$
Sign $1$
Analytic cond. $1545.90$
Root an. cond. $39.3179$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 3·7-s + 6·9-s + 4·13-s − 4·19-s − 9·21-s + 8·23-s − 9·27-s − 6·29-s − 2·31-s − 8·37-s − 12·39-s − 5·41-s + 5·43-s + 3·47-s + 2·49-s + 4·53-s + 12·57-s + 2·59-s + 11·61-s + 18·63-s − 13·67-s − 24·69-s + 2·71-s + 8·73-s + 10·79-s + 9·81-s + ⋯
L(s)  = 1  − 1.73·3-s + 1.13·7-s + 2·9-s + 1.10·13-s − 0.917·19-s − 1.96·21-s + 1.66·23-s − 1.73·27-s − 1.11·29-s − 0.359·31-s − 1.31·37-s − 1.92·39-s − 0.780·41-s + 0.762·43-s + 0.437·47-s + 2/7·49-s + 0.549·53-s + 1.58·57-s + 0.260·59-s + 1.40·61-s + 2.26·63-s − 1.58·67-s − 2.88·69-s + 0.237·71-s + 0.936·73-s + 1.12·79-s + 81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(193600\)    =    \(2^{6} \cdot 5^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(1545.90\)
Root analytic conductor: \(39.3179\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 193600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.692343904\)
\(L(\frac12)\) \(\approx\) \(1.692343904\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good3 \( 1 + p T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 - 11 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
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

−13.02998375949769, −12.52310699952105, −12.07641254211791, −11.50158819391248, −11.22098884061772, −10.87869643661402, −10.52593833051860, −10.10094675404021, −9.198097294052198, −8.879772711097914, −8.363811845874144, −7.691126099245491, −7.217745259646697, −6.690165794121929, −6.343929414296256, −5.615487722856980, −5.347818175466820, −4.925024975430519, −4.339828140162806, −3.845807677393413, −3.226532496824308, −2.123279134475401, −1.689462459756239, −1.031129080430377, −0.4896364332832257, 0.4896364332832257, 1.031129080430377, 1.689462459756239, 2.123279134475401, 3.226532496824308, 3.845807677393413, 4.339828140162806, 4.925024975430519, 5.347818175466820, 5.615487722856980, 6.343929414296256, 6.690165794121929, 7.217745259646697, 7.691126099245491, 8.363811845874144, 8.879772711097914, 9.198097294052198, 10.10094675404021, 10.52593833051860, 10.87869643661402, 11.22098884061772, 11.50158819391248, 12.07641254211791, 12.52310699952105, 13.02998375949769

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