Properties

Label 2-4800-5.4-c1-0-62
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·3-s − 9-s − 4·11-s + 6i·13-s − 6i·17-s + 4·19-s + i·27-s − 2·29-s + 8·31-s + 4i·33-s + 2i·37-s + 6·39-s − 6·41-s − 12i·43-s − 8i·47-s + ⋯
L(s)  = 1  − 0.577i·3-s − 0.333·9-s − 1.20·11-s + 1.66i·13-s − 1.45i·17-s + 0.917·19-s + 0.192i·27-s − 0.371·29-s + 1.43·31-s + 0.696i·33-s + 0.328i·37-s + 0.960·39-s − 0.937·41-s − 1.82i·43-s − 1.16i·47-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: \(1\)
Selberg data: \((2,\ 4800,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 - 6iT - 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 12iT - 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 + 14T + 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
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

−7.68505171874222637609371384280, −7.17901479788646888104313576789, −6.62259967867106113152371797126, −5.61157379274572440855946687444, −5.00983504460379356515178401229, −4.19232013327863941784565934097, −3.01249292175252128478910213895, −2.38621067497179226259211787210, −1.32258944449181823643409938953, 0, 1.40020740604492007621840783555, 2.86035553878473070309564466885, 3.15882638287324621318957699396, 4.31772113101396600190821725962, 5.03694257301407734999652051961, 5.75730889449406377018777755221, 6.26051014659049839744892631401, 7.61845735190994045034698198293, 7.904187680943919217362116464910

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