Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} \cdot 11 \cdot 19 $
Sign $i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.31i·11-s + 2.79·13-s − 4.88i·17-s − 4.35·19-s − 5·25-s + 7·49-s − 8.72i·53-s − 2.72i·59-s − 14.7i·71-s − 17.1·79-s − 1.75i·83-s + 3.27i·89-s − 13.2i·101-s + 17.4·109-s − 20.7i·113-s + ⋯
L(s)  = 1  + 1.00i·11-s + 0.775·13-s − 1.18i·17-s − 1.00·19-s − 25-s + 49-s − 1.19i·53-s − 0.355i·59-s − 1.74i·71-s − 1.92·79-s − 0.192i·83-s + 0.346i·89-s − 1.32i·101-s + 1.67·109-s − 1.94i·113-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 7524 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 7524 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(7524\)    =    \(2^{2} \cdot 3^{2} \cdot 11 \cdot 19\)
\( \varepsilon \)  =  $i$
motivic weight  =  \(1\)
character  :  $\chi_{7524} (2089, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 7524,\ (\ :1/2),\ i)$
$L(1)$  $\approx$  $1.244775186$
$L(\frac12)$  $\approx$  $1.244775186$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;11,\;19\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;3,\;11,\;19\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 - 3.31iT \)
19 \( 1 + 4.35T \)
good5 \( 1 + 5T^{2} \)
7 \( 1 - 7T^{2} \)
13 \( 1 - 2.79T + 13T^{2} \)
17 \( 1 + 4.88iT - 17T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 8.72iT - 53T^{2} \)
59 \( 1 + 2.72iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 14.7iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 17.1T + 79T^{2} \)
83 \( 1 + 1.75iT - 83T^{2} \)
89 \( 1 - 3.27iT - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.61217102575939880045322857028, −7.08183902598457414114434628769, −6.34907938587092690551403175150, −5.65393527705216520274937420835, −4.79583577449806652728662161516, −4.20575996767645532172944864949, −3.37307821788676215199960591400, −2.38119172553117514228671488345, −1.64862461001768516062824284918, −0.32063942211724448659396539156, 1.02000943263608986425193310825, 1.98323766173136007772930153534, 2.95939563736652677395901390828, 3.88122325209651015316477041124, 4.24684519291302496318570751767, 5.49516209442891962852509790612, 5.97843874435572867032659854320, 6.49830505058121752957306682373, 7.43517677280643500312122182905, 8.198987945668227015763917296868

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