Properties

Degree $2$
Conductor $4800$
Sign $-0.894 + 0.447i$
Motivic weight $1$
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: $-0.894 + 0.447i$
Motivic weight: \(1\)
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

\(L(1)\) \(\approx\) \(0.5180487010\)
\(L(\frac12)\) \(\approx\) \(0.5180487010\)
\(L(\frac{3}{2})\) 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

−8.158297965893395875536536566238, −7.19546948894517705814261930376, −6.53632454707921671253133712783, −5.74118921287138040355002067898, −5.25132457113131706421315713230, −4.33627945846281192038255049054, −2.97933740423861091418010559601, −2.59785736426635415508276587317, −1.56798595818064913692494655373, −0.14036159239926709240687008141, 1.24523399701690396111495928999, 2.31470118259016022338550526527, 3.70510808288950872780027081159, 3.94566661095528265927170750141, 4.70127456620026194380752292530, 5.65708101710557455717847389203, 6.56133756697981748068188510870, 7.00973490429193110249063852313, 8.029451383413718376665135192763, 8.455233135705387212861052314482

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