Properties

Label 2-1350-1.1-c1-0-0
Degree $2$
Conductor $1350$
Sign $1$
Analytic cond. $10.7798$
Root an. cond. $3.28326$
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-s + 4-s − 4.35·7-s − 8-s − 4.35·11-s + 4.35·14-s + 16-s − 4·17-s + 6·19-s + 4.35·22-s + 2·23-s − 4.35·28-s + 7·31-s − 32-s + 4·34-s + 8.71·37-s − 6·38-s + 8.71·41-s + 8.71·43-s − 4.35·44-s − 2·46-s − 2·47-s + 12.0·49-s + 3·53-s + 4.35·56-s − 8.71·59-s − 4·61-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 1.64·7-s − 0.353·8-s − 1.31·11-s + 1.16·14-s + 0.250·16-s − 0.970·17-s + 1.37·19-s + 0.929·22-s + 0.417·23-s − 0.823·28-s + 1.25·31-s − 0.176·32-s + 0.685·34-s + 1.43·37-s − 0.973·38-s + 1.36·41-s + 1.32·43-s − 0.657·44-s − 0.294·46-s − 0.291·47-s + 1.71·49-s + 0.412·53-s + 0.582·56-s − 1.13·59-s − 0.512·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(10.7798\)
Root analytic conductor: \(3.28326\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7443064443\)
\(L(\frac12)\) \(\approx\) \(0.7443064443\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 4.35T + 7T^{2} \)
11 \( 1 + 4.35T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 4T + 17T^{2} \)
19 \( 1 - 6T + 19T^{2} \)
23 \( 1 - 2T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 7T + 31T^{2} \)
37 \( 1 - 8.71T + 37T^{2} \)
41 \( 1 - 8.71T + 41T^{2} \)
43 \( 1 - 8.71T + 43T^{2} \)
47 \( 1 + 2T + 47T^{2} \)
53 \( 1 - 3T + 53T^{2} \)
59 \( 1 + 8.71T + 59T^{2} \)
61 \( 1 + 4T + 61T^{2} \)
67 \( 1 + 8.71T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 4.35T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 5T + 83T^{2} \)
89 \( 1 - 8.71T + 89T^{2} \)
97 \( 1 - 4.35T + 97T^{2} \)
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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

−9.501681216921071119435950428531, −9.078532805202948444297079734363, −7.900010689027835410841605275446, −7.31661291654353055780338017549, −6.37285174061349590745702511871, −5.72592797546718418798229420570, −4.47603370267001059250593299994, −3.09309097048369550461470152144, −2.57207605909052160826990951117, −0.66560827769313658255766820234, 0.66560827769313658255766820234, 2.57207605909052160826990951117, 3.09309097048369550461470152144, 4.47603370267001059250593299994, 5.72592797546718418798229420570, 6.37285174061349590745702511871, 7.31661291654353055780338017549, 7.900010689027835410841605275446, 9.078532805202948444297079734363, 9.501681216921071119435950428531

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