Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{3} \cdot 7 $
Sign $-0.755 - 0.654i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73i·5-s + (2 + 1.73i)7-s + 3.46i·11-s + 3.46i·13-s + 1.73i·17-s − 2·19-s + 2.00·25-s − 6·29-s + 2·31-s + (−2.99 + 3.46i)35-s + 37-s + 5.19i·41-s − 1.73i·43-s + 3·47-s + (1.00 + 6.92i)49-s + ⋯
L(s)  = 1  + 0.774i·5-s + (0.755 + 0.654i)7-s + 1.04i·11-s + 0.960i·13-s + 0.420i·17-s − 0.458·19-s + 0.400·25-s − 1.11·29-s + 0.359·31-s + (−0.507 + 0.585i)35-s + 0.164·37-s + 0.811i·41-s − 0.264i·43-s + 0.437·47-s + (0.142 + 0.989i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.755 - 0.654i$
motivic weight  =  \(1\)
character  :  $\chi_{3024} (1567, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 3024,\ (\ :1/2),\ -0.755 - 0.654i)\)
\(L(1)\)  \(\approx\)  \(1.591574404\)
\(L(\frac12)\)  \(\approx\)  \(1.591574404\)
\(L(\frac{3}{2})\)   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 \( 1 + (-2 - 1.73i)T \)
good5 \( 1 - 1.73iT - 5T^{2} \)
11 \( 1 - 3.46iT - 11T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 - 1.73iT - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 - T + 37T^{2} \)
41 \( 1 - 5.19iT - 41T^{2} \)
43 \( 1 + 1.73iT - 43T^{2} \)
47 \( 1 - 3T + 47T^{2} \)
53 \( 1 + 12T + 53T^{2} \)
59 \( 1 + 3T + 59T^{2} \)
61 \( 1 + 10.3iT - 61T^{2} \)
67 \( 1 + 10.3iT - 67T^{2} \)
71 \( 1 + 13.8iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 1.73iT - 79T^{2} \)
83 \( 1 - 9T + 83T^{2} \)
89 \( 1 - 6.92iT - 89T^{2} \)
97 \( 1 - 3.46iT - 97T^{2} \)
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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

−9.174971042401283626012991911753, −8.133164509530775991616523688598, −7.57435919717308253093760719217, −6.67284401114675792687970826959, −6.17456228098049758001621172680, −5.02426904633694317362668116793, −4.46108971332335328465375995974, −3.41495188650871206208726523391, −2.28877149500466420519370970972, −1.69258874758613181219641855651, 0.50244613608157697575908991457, 1.40009139963704119765595218879, 2.74194792590763441872372261865, 3.74763002946318462915919133551, 4.57845852624762671886782080718, 5.34334659418224474882531255050, 5.96806521397015346815143852771, 7.08510638329960507782034790136, 7.79341856962824814822078639654, 8.454369367263264743566368750137

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