Properties

Label 2-5445-1.1-c1-0-126
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.262·2-s − 1.93·4-s − 5-s + 3.19·7-s + 1.03·8-s + 0.262·10-s + 1.11·13-s − 0.837·14-s + 3.59·16-s − 0.0882·17-s + 0.0688·19-s + 1.93·20-s − 6.65·23-s + 25-s − 0.293·26-s − 6.16·28-s + 3.73·29-s − 9.58·31-s − 3.00·32-s + 0.0231·34-s − 3.19·35-s − 8.33·37-s − 0.0180·38-s − 1.03·40-s − 11.6·41-s + 11.8·43-s + 1.74·46-s + ⋯
L(s)  = 1  − 0.185·2-s − 0.965·4-s − 0.447·5-s + 1.20·7-s + 0.364·8-s + 0.0829·10-s + 0.310·13-s − 0.223·14-s + 0.897·16-s − 0.0213·17-s + 0.0157·19-s + 0.431·20-s − 1.38·23-s + 0.200·25-s − 0.0576·26-s − 1.16·28-s + 0.693·29-s − 1.72·31-s − 0.531·32-s + 0.00396·34-s − 0.539·35-s − 1.36·37-s − 0.00292·38-s − 0.163·40-s − 1.82·41-s + 1.80·43-s + 0.257·46-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: \(1\)
Selberg data: \((2,\ 5445,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 \)
5 \( 1 + T \)
11 \( 1 \)
good2 \( 1 + 0.262T + 2T^{2} \)
7 \( 1 - 3.19T + 7T^{2} \)
13 \( 1 - 1.11T + 13T^{2} \)
17 \( 1 + 0.0882T + 17T^{2} \)
19 \( 1 - 0.0688T + 19T^{2} \)
23 \( 1 + 6.65T + 23T^{2} \)
29 \( 1 - 3.73T + 29T^{2} \)
31 \( 1 + 9.58T + 31T^{2} \)
37 \( 1 + 8.33T + 37T^{2} \)
41 \( 1 + 11.6T + 41T^{2} \)
43 \( 1 - 11.8T + 43T^{2} \)
47 \( 1 + 0.908T + 47T^{2} \)
53 \( 1 + 0.872T + 53T^{2} \)
59 \( 1 + 1.83T + 59T^{2} \)
61 \( 1 - 10.0T + 61T^{2} \)
67 \( 1 - 9.53T + 67T^{2} \)
71 \( 1 - 4.66T + 71T^{2} \)
73 \( 1 - 7.16T + 73T^{2} \)
79 \( 1 + 0.791T + 79T^{2} \)
83 \( 1 - 0.247T + 83T^{2} \)
89 \( 1 - 14.5T + 89T^{2} \)
97 \( 1 + 4.09T + 97T^{2} \)
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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

−7.966258356535236508839308183536, −7.36990305595565994111798284139, −6.38526787910295819677089821444, −5.30270528112514281534561338196, −5.02807345888040380917467853993, −3.98925593046077790360933151286, −3.64430492028448604916859428251, −2.16122773625760162792567551441, −1.25579077776624139517339985395, 0, 1.25579077776624139517339985395, 2.16122773625760162792567551441, 3.64430492028448604916859428251, 3.98925593046077790360933151286, 5.02807345888040380917467853993, 5.30270528112514281534561338196, 6.38526787910295819677089821444, 7.36990305595565994111798284139, 7.966258356535236508839308183536

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