Properties

Label 2-4608-8.5-c1-0-30
Degree $2$
Conductor $4608$
Sign $-i$
Analytic cond. $36.7950$
Root an. cond. $6.06589$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.82i·5-s + 4·7-s − 1.41i·11-s − 2.82i·13-s + 4·17-s + 7.07i·19-s − 4·23-s − 3.00·25-s + 8.48i·29-s − 8·31-s + 11.3i·35-s + 2.82i·37-s + 2·41-s + 4.24i·43-s + 9·49-s + ⋯
L(s)  = 1  + 1.26i·5-s + 1.51·7-s − 0.426i·11-s − 0.784i·13-s + 0.970·17-s + 1.62i·19-s − 0.834·23-s − 0.600·25-s + 1.57i·29-s − 1.43·31-s + 1.91i·35-s + 0.464i·37-s + 0.312·41-s + 0.646i·43-s + 1.28·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4608\)    =    \(2^{9} \cdot 3^{2}\)
Sign: $-i$
Analytic conductor: \(36.7950\)
Root analytic conductor: \(6.06589\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4608} (2305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4608,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.207807463\)
\(L(\frac12)\) \(\approx\) \(2.207807463\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 - 2.82iT - 5T^{2} \)
7 \( 1 - 4T + 7T^{2} \)
11 \( 1 + 1.41iT - 11T^{2} \)
13 \( 1 + 2.82iT - 13T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 - 7.07iT - 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 8.48iT - 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 - 2.82iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 - 4.24iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 2.82iT - 53T^{2} \)
59 \( 1 + 4.24iT - 59T^{2} \)
61 \( 1 - 8.48iT - 61T^{2} \)
67 \( 1 + 4.24iT - 67T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 - 4T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 9.89iT - 83T^{2} \)
89 \( 1 - 12T + 89T^{2} \)
97 \( 1 + 4T + 97T^{2} \)
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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

−8.161048167499630293320706591908, −7.85707408947901313955270006254, −7.24084011673961309058733793923, −6.23759932069706447639389250404, −5.59278205061334039699439425990, −4.94967096528404641915530265275, −3.71205497990449823282899726602, −3.27918955886541564057680195097, −2.12474932894076248211621500644, −1.27947935886242528408816348254, 0.63473727976544485320925178187, 1.65741733436685305257799002990, 2.33115270495740586752525859178, 3.90834311019608870711795833723, 4.48786736240964925000394711254, 5.11254444484965708242530121742, 5.65824006542827528276795866023, 6.78719626837114206837172959974, 7.69053922091741220420945393274, 8.033404785342362874929681190587

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