Properties

Label 2-2394-1.1-c1-0-8
Degree $2$
Conductor $2394$
Sign $1$
Analytic cond. $19.1161$
Root an. cond. $4.37220$
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 − 2.19·5-s − 7-s + 8-s − 2.19·10-s + 1.19·11-s − 6.38·13-s − 14-s + 16-s + 4·17-s + 19-s − 2.19·20-s + 1.19·22-s + 8.38·23-s − 0.192·25-s − 6.38·26-s − 28-s + 0.192·29-s + 10·31-s + 32-s + 4·34-s + 2.19·35-s + 4.19·37-s + 38-s − 2.19·40-s − 6.57·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 0.980·5-s − 0.377·7-s + 0.353·8-s − 0.693·10-s + 0.359·11-s − 1.77·13-s − 0.267·14-s + 0.250·16-s + 0.970·17-s + 0.229·19-s − 0.490·20-s + 0.254·22-s + 1.74·23-s − 0.0385·25-s − 1.25·26-s − 0.188·28-s + 0.0357·29-s + 1.79·31-s + 0.176·32-s + 0.685·34-s + 0.370·35-s + 0.689·37-s + 0.162·38-s − 0.346·40-s − 1.02·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2394\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 19\)
Sign: $1$
Analytic conductor: \(19.1161\)
Root analytic conductor: \(4.37220\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2394,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.173457269\)
\(L(\frac12)\) \(\approx\) \(2.173457269\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + T \)
19 \( 1 - T \)
good5 \( 1 + 2.19T + 5T^{2} \)
11 \( 1 - 1.19T + 11T^{2} \)
13 \( 1 + 6.38T + 13T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
23 \( 1 - 8.38T + 23T^{2} \)
29 \( 1 - 0.192T + 29T^{2} \)
31 \( 1 - 10T + 31T^{2} \)
37 \( 1 - 4.19T + 37T^{2} \)
41 \( 1 + 6.57T + 41T^{2} \)
43 \( 1 - 10.1T + 43T^{2} \)
47 \( 1 - 4.80T + 47T^{2} \)
53 \( 1 - 5.19T + 53T^{2} \)
59 \( 1 - 4.57T + 59T^{2} \)
61 \( 1 + 0.807T + 61T^{2} \)
67 \( 1 + 2T + 67T^{2} \)
71 \( 1 - 1.19T + 71T^{2} \)
73 \( 1 + 4.38T + 73T^{2} \)
79 \( 1 - 1.80T + 79T^{2} \)
83 \( 1 + 8T + 83T^{2} \)
89 \( 1 + 2.42T + 89T^{2} \)
97 \( 1 - 8.80T + 97T^{2} \)
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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

−8.966008242149059243322561278381, −7.920761159820696974738027029334, −7.34983570266189996374171002122, −6.77500045012533759583742614552, −5.70231906285146580123494284112, −4.86849504731556748424436028645, −4.21370243555170539445478711553, −3.21991039346502037772994414365, −2.55405266092204699734770178641, −0.850822208850626494616504195114, 0.850822208850626494616504195114, 2.55405266092204699734770178641, 3.21991039346502037772994414365, 4.21370243555170539445478711553, 4.86849504731556748424436028645, 5.70231906285146580123494284112, 6.77500045012533759583742614552, 7.34983570266189996374171002122, 7.920761159820696974738027029334, 8.966008242149059243322561278381

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