Properties

Label 2-6045-1.1-c1-0-88
Degree $2$
Conductor $6045$
Sign $1$
Analytic cond. $48.2695$
Root an. cond. $6.94763$
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  − 0.698·2-s + 3-s − 1.51·4-s + 5-s − 0.698·6-s + 4.99·7-s + 2.45·8-s + 9-s − 0.698·10-s − 1.56·11-s − 1.51·12-s − 13-s − 3.48·14-s + 15-s + 1.31·16-s + 0.300·17-s − 0.698·18-s − 6.71·19-s − 1.51·20-s + 4.99·21-s + 1.09·22-s + 3.87·23-s + 2.45·24-s + 25-s + 0.698·26-s + 27-s − 7.55·28-s + ⋯
L(s)  = 1  − 0.493·2-s + 0.577·3-s − 0.756·4-s + 0.447·5-s − 0.285·6-s + 1.88·7-s + 0.867·8-s + 0.333·9-s − 0.220·10-s − 0.473·11-s − 0.436·12-s − 0.277·13-s − 0.932·14-s + 0.258·15-s + 0.328·16-s + 0.0728·17-s − 0.164·18-s − 1.54·19-s − 0.338·20-s + 1.09·21-s + 0.233·22-s + 0.807·23-s + 0.500·24-s + 0.200·25-s + 0.136·26-s + 0.192·27-s − 1.42·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6045\)    =    \(3 \cdot 5 \cdot 13 \cdot 31\)
Sign: $1$
Analytic conductor: \(48.2695\)
Root analytic conductor: \(6.94763\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6045,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.140789559\)
\(L(\frac12)\) \(\approx\) \(2.140789559\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 - T \)
13 \( 1 + T \)
31 \( 1 - T \)
good2 \( 1 + 0.698T + 2T^{2} \)
7 \( 1 - 4.99T + 7T^{2} \)
11 \( 1 + 1.56T + 11T^{2} \)
17 \( 1 - 0.300T + 17T^{2} \)
19 \( 1 + 6.71T + 19T^{2} \)
23 \( 1 - 3.87T + 23T^{2} \)
29 \( 1 + 3.73T + 29T^{2} \)
37 \( 1 + 5.41T + 37T^{2} \)
41 \( 1 + 2.60T + 41T^{2} \)
43 \( 1 - 10.5T + 43T^{2} \)
47 \( 1 - 5.68T + 47T^{2} \)
53 \( 1 - 6.83T + 53T^{2} \)
59 \( 1 - 12.5T + 59T^{2} \)
61 \( 1 - 13.6T + 61T^{2} \)
67 \( 1 + 8.54T + 67T^{2} \)
71 \( 1 + 12.6T + 71T^{2} \)
73 \( 1 + 3.14T + 73T^{2} \)
79 \( 1 - 6.81T + 79T^{2} \)
83 \( 1 - 12.0T + 83T^{2} \)
89 \( 1 - 4.84T + 89T^{2} \)
97 \( 1 + 7.32T + 97T^{2} \)
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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

−8.192630924861699460911092602596, −7.61032088809800906412114007451, −7.01492764812480644067262555641, −5.70120882061976864873673196112, −5.11151599442579431327673975502, −4.47639830333820609253035422168, −3.84072587833722278446981407206, −2.42196752815811495114885836617, −1.85703480034152668473404898923, −0.846087140066560604539552811700, 0.846087140066560604539552811700, 1.85703480034152668473404898923, 2.42196752815811495114885836617, 3.84072587833722278446981407206, 4.47639830333820609253035422168, 5.11151599442579431327673975502, 5.70120882061976864873673196112, 7.01492764812480644067262555641, 7.61032088809800906412114007451, 8.192630924861699460911092602596

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