Properties

Label 2-4800-5.4-c1-0-66
Degree $2$
Conductor $4800$
Sign $-0.894 + 0.447i$
Analytic cond. $38.3281$
Root an. cond. $6.19097$
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  + i·3-s − 4i·7-s − 9-s − 2i·13-s − 6i·17-s + 4·19-s + 4·21-s i·27-s − 6·29-s − 8·31-s + 2i·37-s + 2·39-s − 6·41-s − 4i·43-s − 9·49-s + ⋯
L(s)  = 1  + 0.577i·3-s − 1.51i·7-s − 0.333·9-s − 0.554i·13-s − 1.45i·17-s + 0.917·19-s + 0.872·21-s − 0.192i·27-s − 1.11·29-s − 1.43·31-s + 0.328i·37-s + 0.320·39-s − 0.937·41-s − 0.609i·43-s − 1.28·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4800\)    =    \(2^{6} \cdot 3 \cdot 5^{2}\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(38.3281\)
Root analytic conductor: \(6.19097\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4800} (3649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4800,\ (\ :1/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7495184164\)
\(L(\frac12)\) \(\approx\) \(0.7495184164\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
5 \( 1 \)
good7 \( 1 + 4iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 + 18T + 89T^{2} \)
97 \( 1 + 2iT - 97T^{2} \)
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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

−7.80235107634565824670375211405, −7.29191628020203835954349274503, −6.75285843438458058155856004519, −5.46589143034714443083993940067, −5.13574751213283209023778066655, −4.06876884228442224896906616215, −3.58157274485400946865577139421, −2.68190670110707957504831908173, −1.26626555704771791504293917164, −0.20269585753409633530984875096, 1.59155952422769810999849494925, 2.12628002976967149965764387077, 3.17936051963050208815382389477, 3.98344400497615501100868244564, 5.25694533812978407052290526532, 5.63252409518542391525213956678, 6.38421865062114281646091783512, 7.11416967298402658014540010980, 7.966263376202387062752261235190, 8.537395137449503963203691087052

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