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Properties

Label 2-3024-252.103-c1-0-28
Degree $2$
Conductor $3024$
Sign $0.303 + 0.952i$
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  + (0.679 − 0.392i)5-s + (−2.49 + 0.891i)7-s + (1.22 + 0.708i)11-s + (−1.50 − 0.868i)13-s + (−5.43 + 3.13i)17-s + (0.736 − 1.27i)19-s + (4.85 − 2.80i)23-s + (−2.19 + 3.79i)25-s + (−3.95 − 6.85i)29-s + 8.41·31-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 + 0.110·47-s + ⋯
L(s)  = 1  + (0.303 − 0.175i)5-s + (−0.941 + 0.337i)7-s + (0.369 + 0.213i)11-s + (−0.417 − 0.240i)13-s + (−1.31 + 0.761i)17-s + (0.168 − 0.292i)19-s + (1.01 − 0.584i)23-s + (−0.438 + 0.759i)25-s + (−0.734 − 1.27i)29-s + 1.51·31-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 + 0.0160·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.303 + 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.303 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.303 + 0.952i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.303 + 0.952i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.244344752\)
\(L(\frac12)\) \(\approx\) \(1.244344752\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.49 - 0.891i)T \)
good5 \( 1 + (-0.679 + 0.392i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.22 - 0.708i)T + (5.5 + 9.52i)T^{2} \)
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 \( 1 + (-4.85 + 2.80i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.95 + 6.85i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 8.41T + 31T^{2} \)
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 \( 1 - 0.110T + 47T^{2} \)
53 \( 1 + (-4.28 - 7.42i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 0.0736T + 59T^{2} \)
61 \( 1 - 1.23iT - 61T^{2} \)
67 \( 1 + 11.8iT - 67T^{2} \)
71 \( 1 + 0.390iT - 71T^{2} \)
73 \( 1 + (-3.70 + 2.13i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 6.00iT - 79T^{2} \)
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

−8.951275371799316862276104123841, −7.76537141203790756423605981465, −6.98483751396575947419021217576, −6.27320725432966259102032159580, −5.66338596862825417475708068703, −4.63011400284270973276828806956, −3.86766301072280169688860745128, −2.77508985490881074818916334500, −2.00211606494816303968815548998, −0.43763823093899098920805852734, 1.04771991370051157038709491006, 2.44659649820509311207478956119, 3.17093896888657420680147869788, 4.17872466838893754743342348960, 4.97518059652656590625867894689, 5.97188675685492782580225250442, 6.78360835912077471331175659890, 7.04282473359067296304494307823, 8.198869326866595112419441311565, 8.973768745580518077293600621977

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