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Properties

Label 2-271332-1.1-c1-0-0
Degree $2$
Conductor $271332$
Sign $1$
Analytic cond. $2166.59$
Root an. cond. $46.5467$
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  + 4·5-s − 4·7-s + 2·11-s − 2·13-s + 8·17-s − 6·19-s + 5·23-s + 11·25-s + 6·29-s + 3·31-s − 16·35-s − 10·37-s − 10·43-s + 9·47-s + 9·49-s + 11·53-s + 8·55-s + 10·59-s + 11·61-s − 8·65-s + 3·67-s + 12·71-s − 13·73-s − 8·77-s + 8·79-s − 10·83-s + 32·85-s + ⋯
L(s)  = 1  + 1.78·5-s − 1.51·7-s + 0.603·11-s − 0.554·13-s + 1.94·17-s − 1.37·19-s + 1.04·23-s + 11/5·25-s + 1.11·29-s + 0.538·31-s − 2.70·35-s − 1.64·37-s − 1.52·43-s + 1.31·47-s + 9/7·49-s + 1.51·53-s + 1.07·55-s + 1.30·59-s + 1.40·61-s − 0.992·65-s + 0.366·67-s + 1.42·71-s − 1.52·73-s − 0.911·77-s + 0.900·79-s − 1.09·83-s + 3.47·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 271332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 271332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(271332\)    =    \(2^{2} \cdot 3^{2} \cdot 7537\)
Sign: $1$
Analytic conductor: \(2166.59\)
Root analytic conductor: \(46.5467\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 271332,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.476279986\)
\(L(\frac12)\) \(\approx\) \(4.476279986\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7537 \( 1 - T \)
good5 \( 1 - 4 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 11 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 11 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 13 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−12.80266853811923, −12.49844634726253, −11.96773764863312, −11.55109770806844, −10.53053614473386, −10.28525194926465, −10.03983308182791, −9.766843396441898, −9.054930943783567, −8.770178990725740, −8.385165646228506, −7.367434667388181, −6.996552819326294, −6.503982156573212, −6.273433448636503, −5.683585628108697, −5.215797054634299, −4.854979236915480, −3.892822139741907, −3.438528127216197, −2.921815953379265, −2.363914649422362, −1.905216647813730, −1.031620585260349, −0.6457963659198617, 0.6457963659198617, 1.031620585260349, 1.905216647813730, 2.363914649422362, 2.921815953379265, 3.438528127216197, 3.892822139741907, 4.854979236915480, 5.215797054634299, 5.683585628108697, 6.273433448636503, 6.503982156573212, 6.996552819326294, 7.367434667388181, 8.385165646228506, 8.770178990725740, 9.054930943783567, 9.766843396441898, 10.03983308182791, 10.28525194926465, 10.53053614473386, 11.55109770806844, 11.96773764863312, 12.49844634726253, 12.80266853811923

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