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Properties

Label 2-3024-63.59-c1-0-22
Degree $2$
Conductor $3024$
Sign $0.922 - 0.386i$
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  + 3.60·5-s + (−0.415 − 2.61i)7-s + 4.96i·11-s + (0.167 + 0.0966i)13-s + (0.257 − 0.446i)17-s + (−1.69 + 0.979i)19-s + 5.49i·23-s + 8.01·25-s + (6.81 − 3.93i)29-s + (−2.78 + 1.60i)31-s + (−1.49 − 9.42i)35-s + (4.09 + 7.10i)37-s + (−3.39 + 5.88i)41-s + (5.07 + 8.78i)43-s + (5.78 − 10.0i)47-s + ⋯
L(s)  = 1  + 1.61·5-s + (−0.157 − 0.987i)7-s + 1.49i·11-s + (0.0464 + 0.0267i)13-s + (0.0625 − 0.108i)17-s + (−0.389 + 0.224i)19-s + 1.14i·23-s + 1.60·25-s + (1.26 − 0.730i)29-s + (−0.500 + 0.289i)31-s + (−0.253 − 1.59i)35-s + (0.673 + 1.16i)37-s + (−0.530 + 0.918i)41-s + (0.773 + 1.33i)43-s + (0.843 − 1.46i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.386i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.922 - 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.922 - 0.386i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.922 - 0.386i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.528174283\)
\(L(\frac12)\) \(\approx\) \(2.528174283\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.415 + 2.61i)T \)
good5 \( 1 - 3.60T + 5T^{2} \)
11 \( 1 - 4.96iT - 11T^{2} \)
13 \( 1 + (-0.167 - 0.0966i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.257 + 0.446i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.69 - 0.979i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 5.49iT - 23T^{2} \)
29 \( 1 + (-6.81 + 3.93i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.78 - 1.60i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.09 - 7.10i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.39 - 5.88i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.07 - 8.78i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.78 + 10.0i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.88 - 3.97i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.80 - 8.32i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.599 - 0.346i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.26 + 2.18i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.8iT - 71T^{2} \)
73 \( 1 + (-5.49 - 3.17i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.99 + 6.91i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.86 + 8.42i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.42 + 7.65i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.49 - 1.43i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.980947459214486265513658635711, −7.909666791778796491773033867597, −7.15447131994541710944699012610, −6.51625571328370915359369297379, −5.81036768224804682350793319810, −4.86372404686385023154114592598, −4.24179061043417711719532926055, −2.98688020143827785615006053930, −2.01024752591436904849692809256, −1.21160731678265253740012689894, 0.867400306657435036998163915029, 2.25982229830537649968392747331, 2.66776321903586846422621924513, 3.85892961118360901488630059574, 5.19194781060989275085691202245, 5.65581651516688478283623454798, 6.24531268474068787090497013487, 6.88405416365987082169315417990, 8.244849991666878322457064630264, 8.828746807835737335845283213435

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