Properties

Label 2-3024-21.20-c1-0-2
Degree $2$
Conductor $3024$
Sign $-0.928 - 0.371i$
Analytic cond. $24.1467$
Root an. cond. $4.91393$
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  + 0.933·5-s + (−2.45 − 0.981i)7-s − 1.75i·11-s + 4.20i·13-s − 0.231·17-s − 5.00i·19-s + 2.61i·23-s − 4.12·25-s + 3.77i·29-s − 0.840i·31-s + (−2.29 − 0.915i)35-s + 3.01·37-s − 8.98·41-s − 2.40·43-s − 1.03·47-s + ⋯
L(s)  = 1  + 0.417·5-s + (−0.928 − 0.371i)7-s − 0.529i·11-s + 1.16i·13-s − 0.0560·17-s − 1.14i·19-s + 0.545i·23-s − 0.825·25-s + 0.700i·29-s − 0.150i·31-s + (−0.387 − 0.154i)35-s + 0.495·37-s − 1.40·41-s − 0.366·43-s − 0.150·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.928 - 0.371i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.928 - 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.928 - 0.371i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.928 - 0.371i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2070896897\)
\(L(\frac12)\) \(\approx\) \(0.2070896897\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.45 + 0.981i)T \)
good5 \( 1 - 0.933T + 5T^{2} \)
11 \( 1 + 1.75iT - 11T^{2} \)
13 \( 1 - 4.20iT - 13T^{2} \)
17 \( 1 + 0.231T + 17T^{2} \)
19 \( 1 + 5.00iT - 19T^{2} \)
23 \( 1 - 2.61iT - 23T^{2} \)
29 \( 1 - 3.77iT - 29T^{2} \)
31 \( 1 + 0.840iT - 31T^{2} \)
37 \( 1 - 3.01T + 37T^{2} \)
41 \( 1 + 8.98T + 41T^{2} \)
43 \( 1 + 2.40T + 43T^{2} \)
47 \( 1 + 1.03T + 47T^{2} \)
53 \( 1 + 9.04iT - 53T^{2} \)
59 \( 1 + 6.77T + 59T^{2} \)
61 \( 1 - 8.93iT - 61T^{2} \)
67 \( 1 + 1.10T + 67T^{2} \)
71 \( 1 - 7.21iT - 71T^{2} \)
73 \( 1 - 3.44iT - 73T^{2} \)
79 \( 1 + 5.29T + 79T^{2} \)
83 \( 1 + 14.8T + 83T^{2} \)
89 \( 1 + 15.0T + 89T^{2} \)
97 \( 1 - 1.63iT - 97T^{2} \)
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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

−9.086653885442738939417346768084, −8.490495067391547219216145507431, −7.33126364452068506638169592087, −6.79679979304859132296294214051, −6.13118939957739342762928641521, −5.28314874491463492682356367052, −4.30611265649556613868448592579, −3.48662582233806518920935140893, −2.58169068481962539640815988543, −1.42690464615715694611721573375, 0.06189150180042783347037869561, 1.65518162571956985751968778646, 2.70475747734011579232518092321, 3.48149514741537629599677167358, 4.47706984796200888221536750175, 5.55405261210483466436744819919, 6.01244164135984558107299588261, 6.78468348854143113566591381868, 7.73488876905249485887412164426, 8.335629199405530184375744817487

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