↑

Properties

Label 2-3784-3784.1307-c0-0-1
Degree $2$
Conductor $3784$
Sign $0.455 + 0.890i$
Analytic cond. $1.88846$
Root an. cond. $1.37421$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.995 − 0.0896i)2-s + (0.829 − 1.10i)3-s + (0.983 + 0.178i)4-s + (−0.924 + 1.02i)6-s + (−0.963 − 0.266i)8-s + (−0.255 − 0.873i)9-s + (−0.599 + 0.800i)11-s + (1.01 − 0.940i)12-s + (0.936 + 0.351i)16-s + (0.521 − 0.428i)17-s + (0.175 + 0.893i)18-s + (1.29 − 0.0386i)19-s + (0.669 − 0.743i)22-s + (−1.09 + 0.845i)24-s + (0.992 + 0.119i)25-s + ⋯
L(s)  = 1  + (−0.995 − 0.0896i)2-s + (0.829 − 1.10i)3-s + (0.983 + 0.178i)4-s + (−0.924 + 1.02i)6-s + (−0.963 − 0.266i)8-s + (−0.255 − 0.873i)9-s + (−0.599 + 0.800i)11-s + (1.01 − 0.940i)12-s + (0.936 + 0.351i)16-s + (0.521 − 0.428i)17-s + (0.175 + 0.893i)18-s + (1.29 − 0.0386i)19-s + (0.669 − 0.743i)22-s + (−1.09 + 0.845i)24-s + (0.992 + 0.119i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.455 + 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.455 + 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3784\)    =    \(2^{3} \cdot 11 \cdot 43\)
Sign: $0.455 + 0.890i$
Analytic conductor: \(1.88846\)
Root analytic conductor: \(1.37421\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3784} (1307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3784,\ (\ :0),\ 0.455 + 0.890i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.150996491\)
\(L(\frac12)\) \(\approx\) \(1.150996491\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.995 + 0.0896i)T \)
11 \( 1 + (0.599 - 0.800i)T \)
43 \( 1 + (-0.0149 - 0.999i)T \)
good3 \( 1 + (-0.829 + 1.10i)T + (-0.280 - 0.959i)T^{2} \)
5 \( 1 + (-0.992 - 0.119i)T^{2} \)
7 \( 1 + (-0.669 + 0.743i)T^{2} \)
13 \( 1 + (0.946 + 0.323i)T^{2} \)
17 \( 1 + (-0.521 + 0.428i)T + (0.193 - 0.981i)T^{2} \)
19 \( 1 + (-1.29 + 0.0386i)T + (0.998 - 0.0598i)T^{2} \)
23 \( 1 + (0.988 - 0.149i)T^{2} \)
29 \( 1 + (0.280 - 0.959i)T^{2} \)
31 \( 1 + (0.646 - 0.762i)T^{2} \)
37 \( 1 + (0.978 + 0.207i)T^{2} \)
41 \( 1 + (0.673 + 0.588i)T + (0.134 + 0.990i)T^{2} \)
47 \( 1 + (0.550 + 0.834i)T^{2} \)
53 \( 1 + (0.599 + 0.800i)T^{2} \)
59 \( 1 + (-1.82 + 0.503i)T + (0.858 - 0.512i)T^{2} \)
61 \( 1 + (0.646 + 0.762i)T^{2} \)
67 \( 1 + (0.172 + 0.117i)T + (0.365 + 0.930i)T^{2} \)
71 \( 1 + (-0.887 + 0.460i)T^{2} \)
73 \( 1 + (1.73 + 0.901i)T + (0.575 + 0.817i)T^{2} \)
79 \( 1 + (0.104 - 0.994i)T^{2} \)
83 \( 1 + (1.14 + 1.62i)T + (-0.337 + 0.941i)T^{2} \)
89 \( 1 + (0.313 - 0.0965i)T + (0.826 - 0.563i)T^{2} \)
97 \( 1 + (0.0206 - 0.0216i)T + (-0.0448 - 0.998i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.493222453881593841654857122346, −7.78267578664899363019394848107, −7.23493567166107932310925450178, −6.91644118522633167306424290982, −5.80041853153408592755977053341, −4.88799866145402834087379547721, −3.37513387150020133167853926905, −2.75191606200721824967521850899, −1.92287179931819538646786443286, −1.01198446968279407674509350655, 1.15034554352431657229398237694, 2.63261460759401186409841997178, 3.14021615959550710643354237970, 3.97773648797930893378380858693, 5.18943758414943541840945449543, 5.75267667573113522946818636631, 6.85158551720093776034630440249, 7.60034426527712644129379779835, 8.419838131292047822179635283515, 8.707582692982551799067701681102

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