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Properties

Label 2-3024-9.7-c1-0-35
Degree $2$
Conductor $3024$
Sign $-0.939 + 0.342i$
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  + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)7-s + (3 − 5.19i)11-s + (−3 − 5.19i)13-s + 2·17-s − 7·19-s + (0.5 + 0.866i)23-s + (2 − 3.46i)25-s + (1 − 1.73i)29-s + (5 + 8.66i)31-s − 0.999·35-s − 6·37-s + (−4 − 6.92i)41-s + (−5 + 8.66i)43-s + (−4 + 6.92i)47-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (0.188 − 0.327i)7-s + (0.904 − 1.56i)11-s + (−0.832 − 1.44i)13-s + 0.485·17-s − 1.60·19-s + (0.104 + 0.180i)23-s + (0.400 − 0.692i)25-s + (0.185 − 0.321i)29-s + (0.898 + 1.55i)31-s − 0.169·35-s − 0.986·37-s + (−0.624 − 1.08i)41-s + (−0.762 + 1.32i)43-s + (−0.583 + 1.01i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.939 + 0.342i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2017, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.939 + 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.046379140\)
\(L(\frac12)\) \(\approx\) \(1.046379140\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3 + 5.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3 + 5.19i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + 7T + 19T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1 + 1.73i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5 - 8.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 + (4 + 6.92i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5 - 8.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4 - 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.5 - 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6 + 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 15T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6 - 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 4T + 89T^{2} \)
97 \( 1 + (-1 + 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.368125679376854296902623333501, −7.899244469002196295053075009512, −6.76306777771102506681160583921, −6.18291423990719461187328971672, −5.24508511591339556192165483470, −4.54093334784865517096649220205, −3.51892137085955809714400187014, −2.84670739736882128859618692850, −1.34285373450897641264155794461, −0.32894135743839250840486763275, 1.73197498306471140245138132329, 2.28364808588270751632386977879, 3.62812930360360602220357648486, 4.46798244111652836455618468523, 4.95445013880922349234186907689, 6.29553034593679976182946176817, 6.84859327435795601663532680916, 7.34248227780917714588148214628, 8.394341078148482039632588681074, 9.057419668867284908765189274790

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