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Properties

Label 2-3024-63.47-c1-0-33
Degree $2$
Conductor $3024$
Sign $-0.663 + 0.747i$
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  + 1.05·5-s + (−1.79 + 1.94i)7-s + 6.24i·11-s + (−0.872 + 0.503i)13-s + (−3.26 − 5.66i)17-s + (−1.73 − 1.00i)19-s − 4.40i·23-s − 3.88·25-s + (−6.12 − 3.53i)29-s + (2.07 + 1.19i)31-s + (−1.89 + 2.04i)35-s + (−3.64 + 6.30i)37-s + (−1.80 − 3.11i)41-s + (−1.60 + 2.78i)43-s + (−1.87 − 3.23i)47-s + ⋯
L(s)  = 1  + 0.472·5-s + (−0.679 + 0.733i)7-s + 1.88i·11-s + (−0.241 + 0.139i)13-s + (−0.792 − 1.37i)17-s + (−0.397 − 0.229i)19-s − 0.917i·23-s − 0.777·25-s + (−1.13 − 0.657i)29-s + (0.372 + 0.214i)31-s + (−0.320 + 0.346i)35-s + (−0.598 + 1.03i)37-s + (−0.281 − 0.487i)41-s + (−0.244 + 0.424i)43-s + (−0.272 − 0.472i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.663 + 0.747i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.663 + 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.663 + 0.747i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.663 + 0.747i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2639963408\)
\(L(\frac12)\) \(\approx\) \(0.2639963408\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.79 - 1.94i)T \)
good5 \( 1 - 1.05T + 5T^{2} \)
11 \( 1 - 6.24iT - 11T^{2} \)
13 \( 1 + (0.872 - 0.503i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.26 + 5.66i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.73 + 1.00i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 4.40iT - 23T^{2} \)
29 \( 1 + (6.12 + 3.53i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.07 - 1.19i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.64 - 6.30i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.80 + 3.11i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.60 - 2.78i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.87 + 3.23i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.02 + 3.47i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.67 + 11.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.10 + 4.10i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.0613 + 0.106i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.37iT - 71T^{2} \)
73 \( 1 + (-14.4 + 8.33i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.43 + 7.67i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.07 - 1.86i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.23 + 3.86i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.960 - 0.554i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.595886602754430247530176561658, −7.58984138327813917307600879325, −6.77273434328368280781988529178, −6.41959578587468300240717081007, −5.18079152758188760158224424837, −4.77188524866854177409605299400, −3.67220069034663077316443838138, −2.33659528799906281643296330322, −2.10978417344305227731235009584, −0.079466137916587433976292748686, 1.26965220783980097790750930107, 2.48135073210252090246872494308, 3.66971503189382025833923980483, 3.93302501699614925505421123008, 5.48263223456471962019162833437, 5.89609769238667154691019833023, 6.64251296477630120200163440974, 7.47403167532338534161339515908, 8.380267001375464836692738295280, 8.900106040466673011712415534112

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