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Properties

Label 2-3024-252.31-c1-0-12
Degree $2$
Conductor $3024$
Sign $0.921 - 0.387i$
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  − 0.784i·5-s + (−2.01 + 1.71i)7-s − 1.41i·11-s + (1.50 + 0.868i)13-s + (−5.43 − 3.13i)17-s + (−0.736 − 1.27i)19-s + 5.60i·23-s + 4.38·25-s + (−3.95 − 6.85i)29-s + (4.20 + 7.29i)31-s + (1.34 + 1.58i)35-s + (3.74 + 6.48i)37-s + (7.19 + 4.15i)41-s + (7.85 − 4.53i)43-s + (0.0550 − 0.0953i)47-s + ⋯
L(s)  = 1  − 0.350i·5-s + (−0.762 + 0.646i)7-s − 0.427i·11-s + (0.417 + 0.240i)13-s + (−1.31 − 0.761i)17-s + (−0.168 − 0.292i)19-s + 1.16i·23-s + 0.876·25-s + (−0.734 − 1.27i)29-s + (0.756 + 1.30i)31-s + (0.227 + 0.267i)35-s + (0.615 + 1.06i)37-s + (1.12 + 0.648i)41-s + (1.19 − 0.691i)43-s + (0.00802 − 0.0139i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 - 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 - 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.921 - 0.387i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1711, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.921 - 0.387i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.469652121\)
\(L(\frac12)\) \(\approx\) \(1.469652121\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.01 - 1.71i)T \)
good5 \( 1 + 0.784iT - 5T^{2} \)
11 \( 1 + 1.41iT - 11T^{2} \)
13 \( 1 + (-1.50 - 0.868i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (5.43 + 3.13i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.736 + 1.27i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 5.60iT - 23T^{2} \)
29 \( 1 + (3.95 + 6.85i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.20 - 7.29i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.74 - 6.48i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-7.19 - 4.15i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-7.85 + 4.53i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.0550 + 0.0953i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.28 + 7.42i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.0368 - 0.0637i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.07 - 0.618i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.2 - 5.93i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.390iT - 71T^{2} \)
73 \( 1 + (-3.70 - 2.13i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.19 - 3.00i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.88 - 13.6i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.15 - 3.55i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.89 - 3.40i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.892517715071807431673623385178, −8.191004945692543502384475483821, −7.14666808032914952331478887349, −6.47942921038917820163842672918, −5.78460768191274557303533644575, −4.93127647484625622127951924826, −4.07390701783845963271581459784, −3.05266266770151180612008511352, −2.27396219212254833820405861436, −0.834965575979892278693288719917, 0.63325223169263537377332451540, 2.10879482307516305961270258364, 3.01006955988909201662267150830, 4.06443443861175911175776115751, 4.51232085541905335044301604201, 5.92635643845770718703459479873, 6.34960573710417070098849947955, 7.17714204249382552540868189949, 7.77375588276619703123998796198, 8.865845902482813045367337410994

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