Properties

Degree $2$
Conductor $3024$
Sign $-0.654 - 0.755i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s + (−2 + 1.73i)7-s − 5.19i·11-s − 3.46i·13-s + 6·17-s + 1.73i·19-s + 5.19i·23-s + 4·25-s − 10.3i·29-s + 5.19i·31-s + (6 − 5.19i)35-s + 37-s + 3·41-s − 10·43-s − 6·47-s + ⋯
L(s)  = 1  − 1.34·5-s + (−0.755 + 0.654i)7-s − 1.56i·11-s − 0.960i·13-s + 1.45·17-s + 0.397i·19-s + 1.08i·23-s + 0.800·25-s − 1.92i·29-s + 0.933i·31-s + (1.01 − 0.878i)35-s + 0.164·37-s + 0.468·41-s − 1.52·43-s − 0.875·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.654 - 0.755i$
Motivic weight: \(1\)
Character: $\chi_{3024} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.654 - 0.755i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2818585877\)
\(L(\frac12)\) \(\approx\) \(0.2818585877\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 - 1.73i)T \)
good5 \( 1 + 3T + 5T^{2} \)
11 \( 1 + 5.19iT - 11T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 - 1.73iT - 19T^{2} \)
23 \( 1 - 5.19iT - 23T^{2} \)
29 \( 1 + 10.3iT - 29T^{2} \)
31 \( 1 - 5.19iT - 31T^{2} \)
37 \( 1 - T + 37T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 + 10T + 43T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 6T + 59T^{2} \)
61 \( 1 - 13.8iT - 61T^{2} \)
67 \( 1 + 2T + 67T^{2} \)
71 \( 1 - 5.19iT - 71T^{2} \)
73 \( 1 - 3.46iT - 73T^{2} \)
79 \( 1 + 14T + 79T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + 9T + 89T^{2} \)
97 \( 1 - 6.92iT - 97T^{2} \)
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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

−8.782123655175865638387133269304, −8.091557451854792137476252105068, −7.81643054515517726610614443296, −6.75475957125653173449070368729, −5.72818608844261016504183430292, −5.48290297729960953092363597728, −4.02357223526393147613956281645, −3.35688742724583589401919427414, −2.88869269192788205398542592838, −1.01932724016355281992637817179, 0.10897078076687579224833851951, 1.54825719913142585626980472826, 2.94561479475134306526077932657, 3.77688240111925188310556803731, 4.40914325042995461081554599157, 5.12754387913780638659933960935, 6.54681280671259737121490475387, 6.99219211832104919156461737014, 7.60277070486980131044180321839, 8.291912165078873982046773746583

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