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Properties

Label 2-3024-63.5-c1-0-21
Degree $2$
Conductor $3024$
Sign $0.542 - 0.840i$
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  + (0.537 + 0.930i)5-s + (1.37 + 2.25i)7-s + (3.55 + 2.04i)11-s + (3.69 + 2.13i)13-s + (0.717 + 1.24i)17-s + (−6.41 − 3.70i)19-s + (5.43 − 3.13i)23-s + (1.92 − 3.32i)25-s + (8.09 − 4.67i)29-s + 6.88i·31-s + (−1.36 + 2.49i)35-s + (−0.453 + 0.785i)37-s + (3.88 − 6.73i)41-s + (6.32 + 10.9i)43-s − 8.43·47-s + ⋯
L(s)  = 1  + (0.240 + 0.416i)5-s + (0.520 + 0.853i)7-s + (1.07 + 0.617i)11-s + (1.02 + 0.592i)13-s + (0.174 + 0.301i)17-s + (−1.47 − 0.850i)19-s + (1.13 − 0.654i)23-s + (0.384 − 0.665i)25-s + (1.50 − 0.868i)29-s + 1.23i·31-s + (−0.230 + 0.421i)35-s + (−0.0745 + 0.129i)37-s + (0.607 − 1.05i)41-s + (0.964 + 1.66i)43-s − 1.23·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.542 - 0.840i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.542 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.542 - 0.840i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2609, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.542 - 0.840i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.402064929\)
\(L(\frac12)\) \(\approx\) \(2.402064929\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.37 - 2.25i)T \)
good5 \( 1 + (-0.537 - 0.930i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.55 - 2.04i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.69 - 2.13i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.717 - 1.24i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.41 + 3.70i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.43 + 3.13i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-8.09 + 4.67i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.88iT - 31T^{2} \)
37 \( 1 + (0.453 - 0.785i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.88 + 6.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.32 - 10.9i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 8.43T + 47T^{2} \)
53 \( 1 + (-1.50 + 0.869i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 6.10T + 59T^{2} \)
61 \( 1 + 2.72iT - 61T^{2} \)
67 \( 1 + 12.0T + 67T^{2} \)
71 \( 1 + 0.783iT - 71T^{2} \)
73 \( 1 + (1.95 - 1.13i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 1.63T + 79T^{2} \)
83 \( 1 + (4.48 + 7.77i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.71 + 2.97i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.05 - 2.91i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.718771094808977913579878193345, −8.431445232357228898306870024508, −7.17775181185320973156163830494, −6.29591952249953109750908404082, −6.25084294288078705003555771885, −4.69024341931469530183868679575, −4.42684167760057344357856996439, −3.08406782139762783988017241206, −2.23281924482041428895085874111, −1.24770163012164910275005014234, 0.910934094600458111155664797619, 1.55164454733765479964809950338, 3.10153547411902678930396948180, 3.90443045424623244942890804500, 4.62400504577565622983843838567, 5.61511954436533031617464122007, 6.29853151708782388845305743217, 7.07972804576953623462908125527, 7.972693588389529316839181901380, 8.643052425308962986254655365449

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