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Properties

Label 2-3024-21.11-c0-0-3
Degree $2$
Conductor $3024$
Sign $0.832 + 0.553i$
Analytic cond. $1.50917$
Root an. cond. $1.22848$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 − 0.707i)5-s + (0.5 − 0.866i)7-s + (1.22 + 0.707i)17-s + (0.5 + 0.866i)19-s + (−1.22 + 0.707i)23-s + (0.499 − 0.866i)25-s + 1.41i·29-s + (0.5 − 0.866i)31-s − 1.41i·35-s − 1.41i·41-s + 43-s + (−1.22 + 0.707i)47-s + (−0.499 − 0.866i)49-s + (−1.22 − 0.707i)53-s + (−0.5 − 0.866i)61-s + ⋯
L(s)  = 1  + (1.22 − 0.707i)5-s + (0.5 − 0.866i)7-s + (1.22 + 0.707i)17-s + (0.5 + 0.866i)19-s + (−1.22 + 0.707i)23-s + (0.499 − 0.866i)25-s + 1.41i·29-s + (0.5 − 0.866i)31-s − 1.41i·35-s − 1.41i·41-s + 43-s + (−1.22 + 0.707i)47-s + (−0.499 − 0.866i)49-s + (−1.22 − 0.707i)53-s + (−0.5 − 0.866i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.832 + 0.553i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :0),\ 0.832 + 0.553i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.696235615\)
\(L(\frac12)\) \(\approx\) \(1.696235615\)
\(L(1)\) 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 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 - 1.41iT - T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + 1.41iT - T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−8.880476730467053130642538462282, −7.970074257901333949327845673210, −7.57092064383756851368275635920, −6.39398263780148151115039535540, −5.69705359462257438400790056812, −5.15651252362655124902344645151, −4.14730706053328914572948200307, −3.32355054762827789746119772661, −1.82875142241042131249999644566, −1.30682075487050978385524933795, 1.47234810638684745760158185021, 2.53970010761624892896023471694, 2.98761077393394687463420709768, 4.46213087679617145068648294754, 5.28717792507574096769966520213, 5.98339037793369038803289710600, 6.49503718419554798353371070627, 7.52816719259517675175450917212, 8.202255005215447709120410153069, 9.095542700133927600392066034411

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