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Properties

Label 2-3024-9.4-c1-0-10
Degree $2$
Conductor $3024$
Sign $0.384 - 0.923i$
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  + (−1.29 + 2.24i)5-s + (−0.5 − 0.866i)7-s + (−2.25 − 3.90i)11-s + (−0.5 + 0.866i)13-s + 0.945·17-s + 4.05·19-s + (0.136 − 0.236i)23-s + (−0.863 − 1.49i)25-s + (1.23 + 2.13i)29-s + (1.16 − 2.01i)31-s + 2.59·35-s + 1.78·37-s + (−3.20 + 5.54i)41-s + (−5.21 − 9.03i)43-s + (6.08 + 10.5i)47-s + ⋯
L(s)  = 1  + (−0.579 + 1.00i)5-s + (−0.188 − 0.327i)7-s + (−0.680 − 1.17i)11-s + (−0.138 + 0.240i)13-s + 0.229·17-s + 0.930·19-s + (0.0284 − 0.0493i)23-s + (−0.172 − 0.299i)25-s + (0.228 + 0.395i)29-s + (0.209 − 0.362i)31-s + 0.438·35-s + 0.292·37-s + (−0.500 + 0.866i)41-s + (−0.795 − 1.37i)43-s + (0.887 + 1.53i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.384 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.384 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.384 - 0.923i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.384 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.255470363\)
\(L(\frac12)\) \(\approx\) \(1.255470363\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.29 - 2.24i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.25 + 3.90i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 0.945T + 17T^{2} \)
19 \( 1 - 4.05T + 19T^{2} \)
23 \( 1 + (-0.136 + 0.236i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.23 - 2.13i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.16 + 2.01i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.78T + 37T^{2} \)
41 \( 1 + (3.20 - 5.54i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.21 + 9.03i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.08 - 10.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 6.27T + 53T^{2} \)
59 \( 1 + (-1.36 + 2.36i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.13 - 1.96i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.90 - 13.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.27T + 71T^{2} \)
73 \( 1 + 1.50T + 73T^{2} \)
79 \( 1 + (-7.35 - 12.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.472 - 0.819i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 14.3T + 89T^{2} \)
97 \( 1 + (-5.74 - 9.95i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.763001827180421620881202161338, −7.984541077151626232897580016566, −7.37521563954284378676847867656, −6.72131093846265036284308165087, −5.87183416806325939587692257884, −5.08030138036105643499190916127, −3.92682316270510290491657835480, −3.24040134576854126308159665591, −2.57448810091211612169585302298, −0.923179596162115159141153386716, 0.50596554448087163993437101572, 1.81554668926605472739836447786, 2.90770674311356464086694725163, 3.92776839640958451897583665166, 4.89807717871471860510735748971, 5.20015059918443407890053922416, 6.29309917232794836068101172038, 7.31777242305725886986648361761, 7.80358728421328715998789239908, 8.572011774009834665519132173640

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