Properties

Degree $2$
Conductor $3024$
Sign $0.926 + 0.375i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.11 + 3.65i)5-s + (−2.19 − 1.47i)7-s + (−0.964 − 1.67i)11-s + (−0.291 − 0.504i)13-s + (−3.61 + 6.25i)17-s + (−2.10 − 3.64i)19-s + (−0.639 + 1.10i)23-s + (−6.41 − 11.1i)25-s + (4.20 − 7.27i)29-s + 0.952·31-s + (10.0 − 4.89i)35-s + (3.03 + 5.25i)37-s + (−1.31 − 2.27i)41-s + (−0.442 + 0.766i)43-s + 5.76·47-s + ⋯
L(s)  = 1  + (−0.944 + 1.63i)5-s + (−0.829 − 0.559i)7-s + (−0.290 − 0.503i)11-s + (−0.0808 − 0.140i)13-s + (−0.875 + 1.51i)17-s + (−0.482 − 0.835i)19-s + (−0.133 + 0.231i)23-s + (−1.28 − 2.22i)25-s + (0.780 − 1.35i)29-s + 0.171·31-s + (1.69 − 0.827i)35-s + (0.498 + 0.863i)37-s + (−0.205 − 0.355i)41-s + (−0.0674 + 0.116i)43-s + 0.840·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 + 0.375i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.926 + 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.926 + 0.375i$
Motivic weight: \(1\)
Character: $\chi_{3024} (2881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.926 + 0.375i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7545203530\)
\(L(\frac12)\) \(\approx\) \(0.7545203530\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.19 + 1.47i)T \)
good5 \( 1 + (2.11 - 3.65i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.964 + 1.67i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.291 + 0.504i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.61 - 6.25i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.10 + 3.64i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.639 - 1.10i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.20 + 7.27i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.952T + 31T^{2} \)
37 \( 1 + (-3.03 - 5.25i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.31 + 2.27i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.442 - 0.766i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 5.76T + 47T^{2} \)
53 \( 1 + (-0.962 + 1.66i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 4.55T + 59T^{2} \)
61 \( 1 + 10.5T + 61T^{2} \)
67 \( 1 - 4.86T + 67T^{2} \)
71 \( 1 - 11.5T + 71T^{2} \)
73 \( 1 + (-0.446 + 0.772i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 11.8T + 79T^{2} \)
83 \( 1 + (5.24 - 9.08i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.87 + 6.71i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.98 - 3.44i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.416790537403368151158895307812, −7.913777520679589802467792472759, −7.07028702588749233010109117815, −6.45704684911103817309038952003, −6.05647637794365759292873540059, −4.45867755329590440063158462472, −3.84987434151922205660116636304, −3.08109183114254872621211214565, −2.34399836133363383388700199864, −0.36049510960503014673433899991, 0.71116136314486116760087329195, 2.09488703629687787545487898588, 3.22357809814272475531892412885, 4.24401535036627780410533124526, 4.82707431147245858098290449781, 5.51885486391965468918888610425, 6.55677676263312037969961891339, 7.38160540208261434446850856386, 8.104921261743238811351680190001, 8.947508556204927793479811558940

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