Properties

Label 2-6048-24.11-c1-0-8
Degree $2$
Conductor $6048$
Sign $-0.482 - 0.875i$
Analytic cond. $48.2935$
Root an. cond. $6.94935$
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  + 0.381·5-s i·7-s − 2.13i·11-s − 6.49i·13-s + 6.97i·17-s − 6.19·19-s + 1.82·23-s − 4.85·25-s + 0.694·29-s + 1.67i·31-s − 0.381i·35-s + 8.06i·37-s − 1.31i·41-s − 7.67·43-s − 6.82·47-s + ⋯
L(s)  = 1  + 0.170·5-s − 0.377i·7-s − 0.643i·11-s − 1.80i·13-s + 1.69i·17-s − 1.42·19-s + 0.380·23-s − 0.970·25-s + 0.129·29-s + 0.301i·31-s − 0.0644i·35-s + 1.32i·37-s − 0.204i·41-s − 1.17·43-s − 0.995·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.482 - 0.875i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.482 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $-0.482 - 0.875i$
Analytic conductor: \(48.2935\)
Root analytic conductor: \(6.94935\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (5615, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6048,\ (\ :1/2),\ -0.482 - 0.875i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5213128330\)
\(L(\frac12)\) \(\approx\) \(0.5213128330\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + iT \)
good5 \( 1 - 0.381T + 5T^{2} \)
11 \( 1 + 2.13iT - 11T^{2} \)
13 \( 1 + 6.49iT - 13T^{2} \)
17 \( 1 - 6.97iT - 17T^{2} \)
19 \( 1 + 6.19T + 19T^{2} \)
23 \( 1 - 1.82T + 23T^{2} \)
29 \( 1 - 0.694T + 29T^{2} \)
31 \( 1 - 1.67iT - 31T^{2} \)
37 \( 1 - 8.06iT - 37T^{2} \)
41 \( 1 + 1.31iT - 41T^{2} \)
43 \( 1 + 7.67T + 43T^{2} \)
47 \( 1 + 6.82T + 47T^{2} \)
53 \( 1 + 0.954T + 53T^{2} \)
59 \( 1 + 12.8iT - 59T^{2} \)
61 \( 1 - 12.4iT - 61T^{2} \)
67 \( 1 + 0.634T + 67T^{2} \)
71 \( 1 + 5.79T + 71T^{2} \)
73 \( 1 - 8.13T + 73T^{2} \)
79 \( 1 - 14.1iT - 79T^{2} \)
83 \( 1 - 3.36iT - 83T^{2} \)
89 \( 1 - 15.9iT - 89T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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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

−8.219332860774598142220758605323, −7.893179045362077662331265013535, −6.72070143145098319105329387874, −6.19919322601900481133559858933, −5.54216212979595990196522182303, −4.73311339060037879291431368018, −3.75059217019639796382523661284, −3.23122573554382312485318815340, −2.13636408269797885033680297265, −1.10618231408343695992888506152, 0.13264207384251671159826389810, 1.81154292538306704923872175775, 2.23805386499072799038433281840, 3.35194050381337937094250117482, 4.44130776527571590509558143265, 4.71746070402366046709730000552, 5.76570152367776408252251251782, 6.53432687492906609463583320476, 7.04223674713090738592598499251, 7.74911273861768013167613850952

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