Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{3} \cdot 7 $
Sign $-0.981 - 0.188i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 2.59i)7-s + 5.19i·13-s + 5.19i·19-s − 5·25-s − 10.3i·31-s − 11·37-s + 8·43-s + (−6.5 − 2.59i)49-s − 15.5i·61-s − 5·67-s + 15.5i·73-s − 17·79-s + (−13.5 − 2.59i)91-s + 5.19i·97-s + 15.5i·103-s + ⋯
L(s)  = 1  + (−0.188 + 0.981i)7-s + 1.44i·13-s + 1.19i·19-s − 25-s − 1.86i·31-s − 1.80·37-s + 1.21·43-s + (−0.928 − 0.371i)49-s − 1.99i·61-s − 0.610·67-s + 1.82i·73-s − 1.91·79-s + (−1.41 − 0.272i)91-s + 0.527i·97-s + 1.53i·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 - 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.981 - 0.188i$
motivic weight  =  \(1\)
character  :  $\chi_{3024} (1889, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 3024,\ (\ :1/2),\ -0.981 - 0.188i)\)
\(L(1)\)  \(\approx\)  \(0.7146996348\)
\(L(\frac12)\)  \(\approx\)  \(0.7146996348\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (0.5 - 2.59i)T \)
good5 \( 1 + 5T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 5.19iT - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 5.19iT - 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 10.3iT - 31T^{2} \)
37 \( 1 + 11T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 15.5iT - 61T^{2} \)
67 \( 1 + 5T + 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 15.5iT - 73T^{2} \)
79 \( 1 + 17T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 5.19iT - 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.165387615172738355956103439369, −8.357330391002004414599934965309, −7.65332087439882210376020394014, −6.71992948916849747923225807544, −6.00743432451277295054161644537, −5.40616425195147901608068888495, −4.29676592407013649218779304059, −3.62752152754473742931656884443, −2.38246657800215322496234636658, −1.69870850497710224139621714426, 0.21912240264534107090953427530, 1.37995240876712510697855080243, 2.80756775759783184638178282180, 3.49768609938151940951436602801, 4.47017830160339085895272895838, 5.27083927316942115429906546531, 6.07516996107891179920091611907, 7.11453679934263347132880038338, 7.41437739294133205338152666427, 8.405559809908038270019324679269

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