Properties

Label 2-5445-1.1-c1-0-108
Degree $2$
Conductor $5445$
Sign $1$
Analytic cond. $43.4785$
Root an. cond. $6.59382$
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  + 2.75·2-s + 5.60·4-s − 5-s − 1.66·7-s + 9.94·8-s − 2.75·10-s + 3.84·13-s − 4.60·14-s + 16.2·16-s + 1.66·17-s − 5.51·19-s − 5.60·20-s + 9.21·23-s + 25-s + 10.6·26-s − 9.36·28-s − 8.85·29-s + 4·31-s + 24.8·32-s + 4.60·34-s + 1.66·35-s + 7.21·37-s − 15.2·38-s − 9.94·40-s + 2.17·41-s + 9.36·43-s + 25.4·46-s + ⋯
L(s)  = 1  + 1.95·2-s + 2.80·4-s − 0.447·5-s − 0.631·7-s + 3.51·8-s − 0.872·10-s + 1.06·13-s − 1.23·14-s + 4.05·16-s + 0.405·17-s − 1.26·19-s − 1.25·20-s + 1.92·23-s + 0.200·25-s + 2.07·26-s − 1.76·28-s − 1.64·29-s + 0.718·31-s + 4.38·32-s + 0.789·34-s + 0.282·35-s + 1.18·37-s − 2.46·38-s − 1.57·40-s + 0.339·41-s + 1.42·43-s + 3.74·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5445\)    =    \(3^{2} \cdot 5 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(43.4785\)
Root analytic conductor: \(6.59382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5445} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5445,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.246720788\)
\(L(\frac12)\) \(\approx\) \(7.246720788\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + T \)
11 \( 1 \)
good2 \( 1 - 2.75T + 2T^{2} \)
7 \( 1 + 1.66T + 7T^{2} \)
13 \( 1 - 3.84T + 13T^{2} \)
17 \( 1 - 1.66T + 17T^{2} \)
19 \( 1 + 5.51T + 19T^{2} \)
23 \( 1 - 9.21T + 23T^{2} \)
29 \( 1 + 8.85T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 7.21T + 37T^{2} \)
41 \( 1 - 2.17T + 41T^{2} \)
43 \( 1 - 9.36T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 3.21T + 53T^{2} \)
59 \( 1 + 9.21T + 59T^{2} \)
61 \( 1 + 3.33T + 61T^{2} \)
67 \( 1 - 1.21T + 67T^{2} \)
71 \( 1 - 9.21T + 71T^{2} \)
73 \( 1 - 7.18T + 73T^{2} \)
79 \( 1 + 5.51T + 79T^{2} \)
83 \( 1 - 0.505T + 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 4.78T + 97T^{2} \)
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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

−7.77709160718913536432195685625, −7.19703851346823969368580062794, −6.37387485153721164225139136551, −6.04123638359797700975869176791, −5.17557012683255485434441367593, −4.39088651173644456845956077993, −3.77884972825964264207014150715, −3.14401645496203703534112197020, −2.37160232600742878116257939697, −1.15217672718412172093330592546, 1.15217672718412172093330592546, 2.37160232600742878116257939697, 3.14401645496203703534112197020, 3.77884972825964264207014150715, 4.39088651173644456845956077993, 5.17557012683255485434441367593, 6.04123638359797700975869176791, 6.37387485153721164225139136551, 7.19703851346823969368580062794, 7.77709160718913536432195685625

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