Properties

Label 2-8043-1.1-c1-0-136
Degree $2$
Conductor $8043$
Sign $1$
Analytic cond. $64.2236$
Root an. cond. $8.01396$
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  + 0.861·2-s + 3-s − 1.25·4-s + 1.99·5-s + 0.861·6-s − 7-s − 2.80·8-s + 9-s + 1.72·10-s − 2.71·11-s − 1.25·12-s + 1.81·13-s − 0.861·14-s + 1.99·15-s + 0.0975·16-s + 6.34·17-s + 0.861·18-s + 7.67·19-s − 2.51·20-s − 21-s − 2.33·22-s − 0.859·23-s − 2.80·24-s − 1.00·25-s + 1.56·26-s + 27-s + 1.25·28-s + ⋯
L(s)  = 1  + 0.609·2-s + 0.577·3-s − 0.628·4-s + 0.893·5-s + 0.351·6-s − 0.377·7-s − 0.992·8-s + 0.333·9-s + 0.544·10-s − 0.817·11-s − 0.363·12-s + 0.504·13-s − 0.230·14-s + 0.515·15-s + 0.0243·16-s + 1.53·17-s + 0.203·18-s + 1.75·19-s − 0.561·20-s − 0.218·21-s − 0.498·22-s − 0.179·23-s − 0.572·24-s − 0.201·25-s + 0.307·26-s + 0.192·27-s + 0.237·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8043\)    =    \(3 \cdot 7 \cdot 383\)
Sign: $1$
Analytic conductor: \(64.2236\)
Root analytic conductor: \(8.01396\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8043,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.304285604\)
\(L(\frac12)\) \(\approx\) \(3.304285604\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 + T \)
383 \( 1 + T \)
good2 \( 1 - 0.861T + 2T^{2} \)
5 \( 1 - 1.99T + 5T^{2} \)
11 \( 1 + 2.71T + 11T^{2} \)
13 \( 1 - 1.81T + 13T^{2} \)
17 \( 1 - 6.34T + 17T^{2} \)
19 \( 1 - 7.67T + 19T^{2} \)
23 \( 1 + 0.859T + 23T^{2} \)
29 \( 1 + 6.37T + 29T^{2} \)
31 \( 1 + 7.89T + 31T^{2} \)
37 \( 1 - 10.7T + 37T^{2} \)
41 \( 1 - 8.17T + 41T^{2} \)
43 \( 1 + 3.29T + 43T^{2} \)
47 \( 1 + 7.88T + 47T^{2} \)
53 \( 1 + 2.23T + 53T^{2} \)
59 \( 1 - 6.19T + 59T^{2} \)
61 \( 1 - 10.7T + 61T^{2} \)
67 \( 1 + 4.71T + 67T^{2} \)
71 \( 1 - 4.36T + 71T^{2} \)
73 \( 1 + 7.55T + 73T^{2} \)
79 \( 1 + 15.0T + 79T^{2} \)
83 \( 1 - 12.0T + 83T^{2} \)
89 \( 1 + 15.5T + 89T^{2} \)
97 \( 1 - 15.0T + 97T^{2} \)
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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

−7.74843674173539984395536974506, −7.32193091573390089282973444922, −6.06759726113481787381161893944, −5.63310974724433486325468181713, −5.24579511597298776935961510471, −4.19286149905362485081812021781, −3.37857392444642459852732905304, −3.00647192090518678593028307643, −1.91361353528452294620896279169, −0.813479761340096923323223308587, 0.813479761340096923323223308587, 1.91361353528452294620896279169, 3.00647192090518678593028307643, 3.37857392444642459852732905304, 4.19286149905362485081812021781, 5.24579511597298776935961510471, 5.63310974724433486325468181713, 6.06759726113481787381161893944, 7.32193091573390089282973444922, 7.74843674173539984395536974506

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