Properties

Label 2-5445-1.1-c1-0-118
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  + 2.34·2-s + 3.51·4-s + 5-s + 3.38·7-s + 3.56·8-s + 2.34·10-s − 1.48·13-s + 7.94·14-s + 1.33·16-s − 3.73·17-s + 3.21·19-s + 3.51·20-s + 2.51·23-s + 25-s − 3.48·26-s + 11.9·28-s + 4.69·29-s + 9.21·31-s − 3.98·32-s − 8.76·34-s + 3.38·35-s − 2.69·37-s + 7.55·38-s + 3.56·40-s + 4.55·41-s − 5.38·43-s + 5.91·46-s + ⋯
L(s)  = 1  + 1.66·2-s + 1.75·4-s + 0.447·5-s + 1.27·7-s + 1.26·8-s + 0.742·10-s − 0.411·13-s + 2.12·14-s + 0.334·16-s − 0.905·17-s + 0.737·19-s + 0.786·20-s + 0.524·23-s + 0.200·25-s − 0.683·26-s + 2.24·28-s + 0.872·29-s + 1.65·31-s − 0.704·32-s − 1.50·34-s + 0.571·35-s − 0.443·37-s + 1.22·38-s + 0.563·40-s + 0.710·41-s − 0.820·43-s + 0.871·46-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

\(L(1)\) \(\approx\) \(7.018568607\)
\(L(\frac12)\) \(\approx\) \(7.018568607\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.34T + 2T^{2} \)
7 \( 1 - 3.38T + 7T^{2} \)
13 \( 1 + 1.48T + 13T^{2} \)
17 \( 1 + 3.73T + 17T^{2} \)
19 \( 1 - 3.21T + 19T^{2} \)
23 \( 1 - 2.51T + 23T^{2} \)
29 \( 1 - 4.69T + 29T^{2} \)
31 \( 1 - 9.21T + 31T^{2} \)
37 \( 1 + 2.69T + 37T^{2} \)
41 \( 1 - 4.55T + 41T^{2} \)
43 \( 1 + 5.38T + 43T^{2} \)
47 \( 1 - 10.5T + 47T^{2} \)
53 \( 1 - 2.51T + 53T^{2} \)
59 \( 1 + 11.9T + 59T^{2} \)
61 \( 1 - 13.6T + 61T^{2} \)
67 \( 1 - 7.83T + 67T^{2} \)
71 \( 1 + 2.51T + 71T^{2} \)
73 \( 1 - 2.96T + 73T^{2} \)
79 \( 1 + 6.76T + 79T^{2} \)
83 \( 1 - 7.55T + 83T^{2} \)
89 \( 1 + 10.8T + 89T^{2} \)
97 \( 1 - 11.0T + 97T^{2} \)
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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

−8.027466234383112760455222679380, −7.13206781362832217428780023468, −6.59036536716413237333469272615, −5.76088084921901612141555113296, −5.05159657051830800521581402642, −4.68198138359579640974366925450, −3.94423903631894487757513789550, −2.82334030833639937402335808574, −2.29272767868299537219173521415, −1.21385618253500898383900398656, 1.21385618253500898383900398656, 2.29272767868299537219173521415, 2.82334030833639937402335808574, 3.94423903631894487757513789550, 4.68198138359579640974366925450, 5.05159657051830800521581402642, 5.76088084921901612141555113296, 6.59036536716413237333469272615, 7.13206781362832217428780023468, 8.027466234383112760455222679380

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