Properties

Degree $2$
Conductor $2304$
Sign $-0.382 + 0.923i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 − 2i)5-s − 4.24i·7-s + (2.82 + 2.82i)11-s + (3 − 3i)13-s + 6·17-s + (1.41 − 1.41i)19-s − 2.82i·23-s + 3i·25-s + (−4 + 4i)29-s + 4.24·31-s + (−8.48 + 8.48i)35-s + (3 + 3i)37-s − 10i·41-s + (4.24 + 4.24i)43-s + 2.82·47-s + ⋯
L(s)  = 1  + (−0.894 − 0.894i)5-s − 1.60i·7-s + (0.852 + 0.852i)11-s + (0.832 − 0.832i)13-s + 1.45·17-s + (0.324 − 0.324i)19-s − 0.589i·23-s + 0.600i·25-s + (−0.742 + 0.742i)29-s + 0.762·31-s + (−1.43 + 1.43i)35-s + (0.493 + 0.493i)37-s − 1.56i·41-s + (0.646 + 0.646i)43-s + 0.412·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $-0.382 + 0.923i$
Motivic weight: \(1\)
Character: $\chi_{2304} (1729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1/2),\ -0.382 + 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.626391263\)
\(L(\frac12)\) \(\approx\) \(1.626391263\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 + 2i)T + 5iT^{2} \)
7 \( 1 + 4.24iT - 7T^{2} \)
11 \( 1 + (-2.82 - 2.82i)T + 11iT^{2} \)
13 \( 1 + (-3 + 3i)T - 13iT^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 + (-1.41 + 1.41i)T - 19iT^{2} \)
23 \( 1 + 2.82iT - 23T^{2} \)
29 \( 1 + (4 - 4i)T - 29iT^{2} \)
31 \( 1 - 4.24T + 31T^{2} \)
37 \( 1 + (-3 - 3i)T + 37iT^{2} \)
41 \( 1 + 10iT - 41T^{2} \)
43 \( 1 + (-4.24 - 4.24i)T + 43iT^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 + (4 + 4i)T + 53iT^{2} \)
59 \( 1 + 59iT^{2} \)
61 \( 1 + (-3 + 3i)T - 61iT^{2} \)
67 \( 1 + (2.82 - 2.82i)T - 67iT^{2} \)
71 \( 1 - 2.82iT - 71T^{2} \)
73 \( 1 + 16iT - 73T^{2} \)
79 \( 1 + 4.24T + 79T^{2} \)
83 \( 1 + (11.3 - 11.3i)T - 83iT^{2} \)
89 \( 1 + 14iT - 89T^{2} \)
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

−8.648229544677480520196476942816, −7.87058978211184551367311055249, −7.42157575355835099623711553998, −6.61076662830855751230710230030, −5.48051986940857899689547142481, −4.53037231747636614780756066953, −3.97848449908377327645617499458, −3.25648033433620113852056975679, −1.33475434791271000162803324306, −0.68095069863424648489079581465, 1.36442122673214005782188696933, 2.73742679783199083848775227706, 3.44187378014585571285971055976, 4.19038354001089997481501250416, 5.67748616431684756353462742180, 5.96017808759052343894584113349, 6.89992027941922379012705046960, 7.83123019165525030910594958723, 8.412905344227373436406251453878, 9.221636185498458257202756041682

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