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Properties

Label 2-3024-9.7-c1-0-29
Degree $2$
Conductor $3024$
Sign $0.173 + 0.984i$
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  + (1.26 + 2.19i)5-s + (0.5 − 0.866i)7-s + (−0.233 + 0.405i)11-s + (−2.91 − 5.04i)13-s − 3.87·17-s + 2.18·19-s + (0.0530 + 0.0918i)23-s + (−0.705 + 1.22i)25-s + (4.39 − 7.60i)29-s + (−3.84 − 6.65i)31-s + 2.53·35-s − 7.68·37-s + (−1.11 − 1.92i)41-s + (0.613 − 1.06i)43-s + (2.66 − 4.61i)47-s + ⋯
L(s)  = 1  + (0.566 + 0.980i)5-s + (0.188 − 0.327i)7-s + (−0.0705 + 0.122i)11-s + (−0.807 − 1.39i)13-s − 0.940·17-s + 0.501·19-s + (0.0110 + 0.0191i)23-s + (−0.141 + 0.244i)25-s + (0.815 − 1.41i)29-s + (−0.689 − 1.19i)31-s + 0.428·35-s − 1.26·37-s + (−0.173 − 0.301i)41-s + (0.0935 − 0.162i)43-s + (0.388 − 0.673i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.173 + 0.984i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2017, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.386655568\)
\(L(\frac12)\) \(\approx\) \(1.386655568\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (-1.26 - 2.19i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.233 - 0.405i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.91 + 5.04i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 3.87T + 17T^{2} \)
19 \( 1 - 2.18T + 19T^{2} \)
23 \( 1 + (-0.0530 - 0.0918i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.39 + 7.60i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.84 + 6.65i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 7.68T + 37T^{2} \)
41 \( 1 + (1.11 + 1.92i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.613 + 1.06i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.66 + 4.61i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 0.716T + 53T^{2} \)
59 \( 1 + (0.368 + 0.637i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.479 - 0.829i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.81 + 8.34i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.2T + 71T^{2} \)
73 \( 1 + 10.2T + 73T^{2} \)
79 \( 1 + (6.31 - 10.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.36 + 2.36i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 8.11T + 89T^{2} \)
97 \( 1 + (-6.80 + 11.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.476152160792746254591937905805, −7.66472874283862694543040543409, −7.11737667243853419484661959821, −6.30335735558709504019385435455, −5.56469769596748632177282862734, −4.73989072564186879641686342765, −3.69333962510757171965602765239, −2.73345235586049779668365882675, −2.08717260733584341751018240730, −0.41994142237343550909924097011, 1.33965778957413149720389951765, 2.09219004006189121692826232082, 3.24183432595968349284447898071, 4.52589104318284472520102362456, 4.93483038074410307725698429805, 5.69849336734780345032934259118, 6.74531756735085853826309342634, 7.22177715535040945076444494269, 8.433946550120426885057858771451, 8.980655471547831265841611625395

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