Properties

Label 2-945-1.1-c1-0-27
Degree $2$
Conductor $945$
Sign $-1$
Analytic cond. $7.54586$
Root an. cond. $2.74697$
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  + 0.618·2-s − 1.61·4-s + 5-s − 7-s − 2.23·8-s + 0.618·10-s + 3.47·11-s − 6.09·13-s − 0.618·14-s + 1.85·16-s − 6.61·17-s + 0.381·19-s − 1.61·20-s + 2.14·22-s − 4.38·23-s + 25-s − 3.76·26-s + 1.61·28-s − 2.85·29-s + 3·31-s + 5.61·32-s − 4.09·34-s − 35-s − 3·37-s + 0.236·38-s − 2.23·40-s + 0.618·41-s + ⋯
L(s)  = 1  + 0.437·2-s − 0.809·4-s + 0.447·5-s − 0.377·7-s − 0.790·8-s + 0.195·10-s + 1.04·11-s − 1.68·13-s − 0.165·14-s + 0.463·16-s − 1.60·17-s + 0.0876·19-s − 0.361·20-s + 0.457·22-s − 0.913·23-s + 0.200·25-s − 0.738·26-s + 0.305·28-s − 0.529·29-s + 0.538·31-s + 0.993·32-s − 0.701·34-s − 0.169·35-s − 0.493·37-s + 0.0382·38-s − 0.353·40-s + 0.0965·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $-1$
Analytic conductor: \(7.54586\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 945,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 + T \)
good2 \( 1 - 0.618T + 2T^{2} \)
11 \( 1 - 3.47T + 11T^{2} \)
13 \( 1 + 6.09T + 13T^{2} \)
17 \( 1 + 6.61T + 17T^{2} \)
19 \( 1 - 0.381T + 19T^{2} \)
23 \( 1 + 4.38T + 23T^{2} \)
29 \( 1 + 2.85T + 29T^{2} \)
31 \( 1 - 3T + 31T^{2} \)
37 \( 1 + 3T + 37T^{2} \)
41 \( 1 - 0.618T + 41T^{2} \)
43 \( 1 + 7.76T + 43T^{2} \)
47 \( 1 + 10.7T + 47T^{2} \)
53 \( 1 + 6.61T + 53T^{2} \)
59 \( 1 + 10.7T + 59T^{2} \)
61 \( 1 - 1.14T + 61T^{2} \)
67 \( 1 - 12.7T + 67T^{2} \)
71 \( 1 + 8.85T + 71T^{2} \)
73 \( 1 + 2.52T + 73T^{2} \)
79 \( 1 - 10.7T + 79T^{2} \)
83 \( 1 - 14.4T + 83T^{2} \)
89 \( 1 + 13.4T + 89T^{2} \)
97 \( 1 - 16.0T + 97T^{2} \)
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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

−9.550660287951508541321813423167, −9.044804327219182083733636154064, −8.041983683939172592327563607475, −6.82249740246230560233059689989, −6.19626267906589685771942440380, −5.02573903522813994495500054452, −4.41379801848985815164581208006, −3.31625884416302874363695905680, −2.03468494786555335800175458069, 0, 2.03468494786555335800175458069, 3.31625884416302874363695905680, 4.41379801848985815164581208006, 5.02573903522813994495500054452, 6.19626267906589685771942440380, 6.82249740246230560233059689989, 8.041983683939172592327563607475, 9.044804327219182083733636154064, 9.550660287951508541321813423167

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