Properties

Degree $2$
Conductor $3024$
Sign $-0.188 - 0.981i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 2.59i)7-s + 5.19i·13-s + 7·19-s + 5·25-s − 4·31-s − 11·37-s − 10.3i·43-s + (−6.5 + 2.59i)49-s + 15.5i·61-s + 15.5i·67-s + 15.5i·73-s − 5.19i·79-s + (−13.5 + 2.59i)91-s + 5.19i·97-s − 13·103-s + ⋯
L(s)  = 1  + (0.188 + 0.981i)7-s + 1.44i·13-s + 1.60·19-s + 25-s − 0.718·31-s − 1.80·37-s − 1.58i·43-s + (−0.928 + 0.371i)49-s + 1.99i·61-s + 1.90i·67-s + 1.82i·73-s − 0.584i·79-s + (−1.41 + 0.272i)91-s + 0.527i·97-s − 1.28·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.188 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.188 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.188 - 0.981i$
Motivic weight: \(1\)
Character: $\chi_{3024} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.188 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.629162861\)
\(L(\frac12)\) \(\approx\) \(1.629162861\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 2.59i)T \)
good5 \( 1 - 5T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 5.19iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 7T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + 11T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 10.3iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 15.5iT - 61T^{2} \)
67 \( 1 - 15.5iT - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 15.5iT - 73T^{2} \)
79 \( 1 + 5.19iT - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 5.19iT - 97T^{2} \)
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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

−8.906963346491445421921039671331, −8.421894219359877368881252280662, −7.17612776508977148785431657613, −6.91927548056159077734480124059, −5.67859675803281510776382515499, −5.26563074865198188127090752944, −4.26688041801989086698467950158, −3.29950634425457247007850170390, −2.32695934934257295661406682749, −1.36466139652577772071054600575, 0.53776549503771593449730257695, 1.57283632837342061320247865016, 3.10872546596083367057783420229, 3.50960176488685883175680390551, 4.81260312517834832111990639279, 5.25961503727527862990879467740, 6.28782117467784757962816402890, 7.17005599180642840661946160726, 7.71785976537577710025475802949, 8.348423686020990509799095446587

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