Properties

Degree $2$
Conductor $3024$
Sign $-0.0477 - 0.998i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + 1.73i)5-s + (2.5 − 0.866i)7-s + (−2 + 3.46i)11-s + (−1.5 + 2.59i)13-s + (3.5 + 6.06i)17-s + (2.5 − 4.33i)19-s + (−2 − 3.46i)23-s + (0.500 − 0.866i)25-s + (−0.5 − 0.866i)29-s + 3·31-s + (4 + 3.46i)35-s + (−5.5 + 9.52i)37-s + (−4.5 + 7.79i)41-s + (2.5 + 4.33i)43-s + 3·47-s + ⋯
L(s)  = 1  + (0.447 + 0.774i)5-s + (0.944 − 0.327i)7-s + (−0.603 + 1.04i)11-s + (−0.416 + 0.720i)13-s + (0.848 + 1.47i)17-s + (0.573 − 0.993i)19-s + (−0.417 − 0.722i)23-s + (0.100 − 0.173i)25-s + (−0.0928 − 0.160i)29-s + 0.538·31-s + (0.676 + 0.585i)35-s + (−0.904 + 1.56i)37-s + (−0.702 + 1.21i)41-s + (0.381 + 0.660i)43-s + 0.437·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0477 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0477 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.0477 - 0.998i$
Motivic weight: \(1\)
Character: $\chi_{3024} (2305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.0477 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.983240564\)
\(L(\frac12)\) \(\approx\) \(1.983240564\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 + 0.866i)T \)
good5 \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.5 - 2.59i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.5 - 6.06i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3T + 31T^{2} \)
37 \( 1 + (5.5 - 9.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.5 - 4.33i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 3T + 47T^{2} \)
53 \( 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 7T + 59T^{2} \)
61 \( 1 - 3T + 61T^{2} \)
67 \( 1 + 13T + 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (3.5 + 6.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 9T + 79T^{2} \)
83 \( 1 + (0.5 + 0.866i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-7.5 + 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.5 - 14.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.863896888237209275614346742879, −8.023665801355936160892343256101, −7.45683889328566748107510168450, −6.68110331953843445916008562336, −6.00339929482713966264965082087, −4.81478982297972312509778081544, −4.51078543401291395479326039807, −3.19258075270864056401353634079, −2.27211434018871427279312225932, −1.43045237854296266171254502605, 0.62795092053860399159279724238, 1.69032918082396636449036989377, 2.80539567094034626233425533875, 3.70704848426950133069534321467, 5.02450131863824275262345127384, 5.42001097065326163696903606582, 5.81519262853371770136578059810, 7.42200102336156823653123013723, 7.67263848261902962408318831021, 8.633031551396144018726400153143

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