Properties

Degree $2$
Conductor $3024$
Sign $-0.990 + 0.135i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.18i·5-s + (−1 + 2.44i)7-s + 5.26i·11-s + 1.01i·13-s + 3.08i·17-s − 3.24·19-s + 0.903i·23-s + 0.242·25-s − 5.34·29-s − 5.24·31-s + (−5.34 − 2.18i)35-s + 37-s − 8.35i·41-s + 4.47i·43-s + 12.8·47-s + ⋯
L(s)  = 1  + 0.975i·5-s + (−0.377 + 0.925i)7-s + 1.58i·11-s + 0.281i·13-s + 0.748i·17-s − 0.743·19-s + 0.188i·23-s + 0.0485·25-s − 0.992·29-s − 0.941·31-s + (−0.903 − 0.368i)35-s + 0.164·37-s − 1.30i·41-s + 0.682i·43-s + 1.88·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.135i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.990 + 0.135i$
Motivic weight: \(1\)
Character: $\chi_{3024} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.990 + 0.135i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.056421243\)
\(L(\frac12)\) \(\approx\) \(1.056421243\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 2.44i)T \)
good5 \( 1 - 2.18iT - 5T^{2} \)
11 \( 1 - 5.26iT - 11T^{2} \)
13 \( 1 - 1.01iT - 13T^{2} \)
17 \( 1 - 3.08iT - 17T^{2} \)
19 \( 1 + 3.24T + 19T^{2} \)
23 \( 1 - 0.903iT - 23T^{2} \)
29 \( 1 + 5.34T + 29T^{2} \)
31 \( 1 + 5.24T + 31T^{2} \)
37 \( 1 - T + 37T^{2} \)
41 \( 1 + 8.35iT - 41T^{2} \)
43 \( 1 - 4.47iT - 43T^{2} \)
47 \( 1 - 12.8T + 47T^{2} \)
53 \( 1 - 7.55T + 53T^{2} \)
59 \( 1 - 5.34T + 59T^{2} \)
61 \( 1 + 3.46iT - 61T^{2} \)
67 \( 1 + 3.88iT - 67T^{2} \)
71 \( 1 - 2.18iT - 71T^{2} \)
73 \( 1 + 14.2iT - 73T^{2} \)
79 \( 1 - 9.37iT - 79T^{2} \)
83 \( 1 + 12.8T + 83T^{2} \)
89 \( 1 - 3.98iT - 89T^{2} \)
97 \( 1 + 6.33iT - 97T^{2} \)
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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

−9.181825791196771875681549029426, −8.441417585618353460950348315017, −7.33187087634059452897457792280, −7.00488849468532554923180409912, −6.10941982015573886252762869261, −5.43696714736446552976802940682, −4.34059997258093732025911837879, −3.56610683286212394242794853068, −2.42297012342198197177690290857, −1.92020394732292205999879336740, 0.35090222149286493280818778992, 1.15199331182034683631104532574, 2.65888955610607990746564755312, 3.66459551448151719291759094907, 4.31170161848915113701371530606, 5.34959894194110770628168610300, 5.90076204859491296607288729864, 6.89981631347199581240713398842, 7.59611247421757075091600102566, 8.519073450531579383753839744879

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