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Properties

Label 2-4320-9.7-c1-0-22
Degree $2$
Conductor $4320$
Sign $0.999 - 0.0258i$
Analytic cond. $34.4953$
Root an. cond. $5.87327$
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  + (−0.5 − 0.866i)5-s + (−1.69 + 2.93i)7-s + (1.84 − 3.19i)11-s + (0.867 + 1.50i)13-s + 5.93·17-s − 5.10·19-s + (−2.43 − 4.21i)23-s + (−0.499 + 0.866i)25-s + (0.600 − 1.03i)29-s + (2.28 + 3.95i)31-s + 3.39·35-s − 3.14·37-s + (4.32 + 7.48i)41-s + (3.55 − 6.14i)43-s + (1.95 − 3.38i)47-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (−0.641 + 1.11i)7-s + (0.556 − 0.963i)11-s + (0.240 + 0.416i)13-s + 1.43·17-s − 1.17·19-s + (−0.507 − 0.878i)23-s + (−0.0999 + 0.173i)25-s + (0.111 − 0.193i)29-s + (0.410 + 0.710i)31-s + 0.573·35-s − 0.517·37-s + (0.674 + 1.16i)41-s + (0.541 − 0.937i)43-s + (0.284 − 0.493i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4320\)    =    \(2^{5} \cdot 3^{3} \cdot 5\)
Sign: $0.999 - 0.0258i$
Analytic conductor: \(34.4953\)
Root analytic conductor: \(5.87327\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4320} (2881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4320,\ (\ :1/2),\ 0.999 - 0.0258i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.641547805\)
\(L(\frac12)\) \(\approx\) \(1.641547805\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (0.5 + 0.866i)T \)
good7 \( 1 + (1.69 - 2.93i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.84 + 3.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.867 - 1.50i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 5.93T + 17T^{2} \)
19 \( 1 + 5.10T + 19T^{2} \)
23 \( 1 + (2.43 + 4.21i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.600 + 1.03i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.28 - 3.95i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 3.14T + 37T^{2} \)
41 \( 1 + (-4.32 - 7.48i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.55 + 6.14i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.95 + 3.38i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 8.25T + 53T^{2} \)
59 \( 1 + (1.39 + 2.41i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.0729 + 0.126i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.54 - 6.13i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.110T + 71T^{2} \)
73 \( 1 - 15.3T + 73T^{2} \)
79 \( 1 + (2.39 - 4.14i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.38 + 14.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 2.37T + 89T^{2} \)
97 \( 1 + (-3.76 + 6.51i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.504387533836235390305248364715, −7.87550240644066537332762785360, −6.69576672417936586553697081178, −6.15826765779846677263242014960, −5.59548738508412125367205061877, −4.63154224796790359310955154383, −3.71816392694606299791844608957, −3.00713634886232641413585910448, −1.98540607820550218174554435665, −0.72088128472056082450428819521, 0.72769572984088505804438300861, 1.89119449500653670806766744525, 3.10885854685490637208138134566, 3.83700195926462353158139019805, 4.36930767301171326953122487321, 5.50622524213692537911727289135, 6.33881017195499950841878607758, 6.92328448216078996205413404088, 7.68031656224319200239734707029, 8.071469927007663556624399549012

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