Properties

Label 2-440e2-1.1-c1-0-117
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 + 7·17-s + 3·19-s − 9·21-s + 9·27-s + 7·29-s − 3·31-s − 9·37-s + 12·39-s + 4·41-s − 6·43-s − 12·47-s + 2·49-s + 21·51-s + 7·53-s + 9·57-s + 12·59-s + 61-s − 18·63-s + 12·67-s − 9·71-s + 2·73-s − 12·79-s + 9·81-s + ⋯
L(s)  = 1  + 1.73·3-s − 1.13·7-s + 2·9-s + 1.10·13-s + 1.69·17-s + 0.688·19-s − 1.96·21-s + 1.73·27-s + 1.29·29-s − 0.538·31-s − 1.47·37-s + 1.92·39-s + 0.624·41-s − 0.914·43-s − 1.75·47-s + 2/7·49-s + 2.94·51-s + 0.961·53-s + 1.19·57-s + 1.56·59-s + 0.128·61-s − 2.26·63-s + 1.46·67-s − 1.06·71-s + 0.234·73-s − 1.35·79-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: $\chi_{193600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 193600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.092091020\)
\(L(\frac12)\) \(\approx\) \(6.092091020\)
\(L(\frac{3}{2})\) 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 \( 1 - 7 T + p T^{2} \)
19 \( 1 - 3 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + 9 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 7 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 7 T + p T^{2} \)
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

−13.19732374566483, −12.68202517439030, −12.47715364047610, −11.69439539971289, −11.32335204475158, −10.37446205087506, −10.04874152716741, −9.846852723594185, −9.311941864685439, −8.657798611440748, −8.461672027822852, −7.995929995972153, −7.406459189842694, −6.833685549778263, −6.626947157084697, −5.675716669050276, −5.432035510156266, −4.529484802468600, −3.828661796687210, −3.520328332964465, −3.053298354247248, −2.800348927894616, −1.819894757277209, −1.367198504532630, −0.6329598523727472, 0.6329598523727472, 1.367198504532630, 1.819894757277209, 2.800348927894616, 3.053298354247248, 3.520328332964465, 3.828661796687210, 4.529484802468600, 5.432035510156266, 5.675716669050276, 6.626947157084697, 6.833685549778263, 7.406459189842694, 7.995929995972153, 8.461672027822852, 8.657798611440748, 9.311941864685439, 9.846852723594185, 10.04874152716741, 10.37446205087506, 11.32335204475158, 11.69439539971289, 12.47715364047610, 12.68202517439030, 13.19732374566483

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