Properties

Label 2-1968-1.1-c1-0-24
Degree $2$
Conductor $1968$
Sign $1$
Analytic cond. $15.7145$
Root an. cond. $3.96415$
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  + 3-s + 0.853·5-s + 3.83·7-s + 9-s + 3.34·11-s + 3.14·13-s + 0.853·15-s + 3.63·17-s − 1.14·19-s + 3.83·21-s + 2.85·23-s − 4.27·25-s + 27-s − 8.02·29-s − 9.86·31-s + 3.34·33-s + 3.27·35-s + 8.19·37-s + 3.14·39-s + 41-s − 11.7·43-s + 0.853·45-s − 8.32·47-s + 7.68·49-s + 3.63·51-s + 1.60·53-s + 2.85·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.381·5-s + 1.44·7-s + 0.333·9-s + 1.00·11-s + 0.872·13-s + 0.220·15-s + 0.881·17-s − 0.262·19-s + 0.836·21-s + 0.595·23-s − 0.854·25-s + 0.192·27-s − 1.49·29-s − 1.77·31-s + 0.581·33-s + 0.552·35-s + 1.34·37-s + 0.503·39-s + 0.156·41-s − 1.79·43-s + 0.127·45-s − 1.21·47-s + 1.09·49-s + 0.509·51-s + 0.220·53-s + 0.384·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1968\)    =    \(2^{4} \cdot 3 \cdot 41\)
Sign: $1$
Analytic conductor: \(15.7145\)
Root analytic conductor: \(3.96415\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1968,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.065194600\)
\(L(\frac12)\) \(\approx\) \(3.065194600\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
41 \( 1 - T \)
good5 \( 1 - 0.853T + 5T^{2} \)
7 \( 1 - 3.83T + 7T^{2} \)
11 \( 1 - 3.34T + 11T^{2} \)
13 \( 1 - 3.14T + 13T^{2} \)
17 \( 1 - 3.63T + 17T^{2} \)
19 \( 1 + 1.14T + 19T^{2} \)
23 \( 1 - 2.85T + 23T^{2} \)
29 \( 1 + 8.02T + 29T^{2} \)
31 \( 1 + 9.86T + 31T^{2} \)
37 \( 1 - 8.19T + 37T^{2} \)
43 \( 1 + 11.7T + 43T^{2} \)
47 \( 1 + 8.32T + 47T^{2} \)
53 \( 1 - 1.60T + 53T^{2} \)
59 \( 1 - 11.6T + 59T^{2} \)
61 \( 1 + 4.19T + 61T^{2} \)
67 \( 1 + 6.10T + 67T^{2} \)
71 \( 1 - 8.65T + 71T^{2} \)
73 \( 1 + 10.1T + 73T^{2} \)
79 \( 1 - 3.60T + 79T^{2} \)
83 \( 1 - 10.1T + 83T^{2} \)
89 \( 1 + 3.37T + 89T^{2} \)
97 \( 1 + 7.53T + 97T^{2} \)
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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

−9.166877800663270404609084892064, −8.351900259277464373228833241614, −7.79211639937984328309381393487, −6.94298732162077274811567086031, −5.90211426339496441221757636475, −5.17957646422206796897550574483, −4.11026476465645598209038611595, −3.43578217571288695670749784037, −1.94741979671895666719330007466, −1.37936234312947098875478348087, 1.37936234312947098875478348087, 1.94741979671895666719330007466, 3.43578217571288695670749784037, 4.11026476465645598209038611595, 5.17957646422206796897550574483, 5.90211426339496441221757636475, 6.94298732162077274811567086031, 7.79211639937984328309381393487, 8.351900259277464373228833241614, 9.166877800663270404609084892064

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