Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{3} \cdot 7 $
Sign $0.580 - 0.814i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (1.5 + 2.59i)5-s + (2 − 1.73i)7-s + (1.5 − 2.59i)11-s + (0.5 − 0.866i)13-s + (1.5 + 2.59i)17-s + (−3.5 + 6.06i)19-s + (4.5 + 7.79i)23-s + (−2 + 3.46i)25-s + (1.5 + 2.59i)29-s − 8·31-s + (7.5 + 2.59i)35-s + (0.5 − 0.866i)37-s + (1.5 − 2.59i)41-s + (−0.5 − 0.866i)43-s + (1.00 − 6.92i)49-s + ⋯
L(s)  = 1  + (0.670 + 1.16i)5-s + (0.755 − 0.654i)7-s + (0.452 − 0.783i)11-s + (0.138 − 0.240i)13-s + (0.363 + 0.630i)17-s + (−0.802 + 1.39i)19-s + (0.938 + 1.62i)23-s + (−0.400 + 0.692i)25-s + (0.278 + 0.482i)29-s − 1.43·31-s + (1.26 + 0.439i)35-s + (0.0821 − 0.142i)37-s + (0.234 − 0.405i)41-s + (−0.0762 − 0.132i)43-s + (0.142 − 0.989i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.580 - 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.580 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.580 - 0.814i$
motivic weight  =  \(1\)
character  :  $\chi_{3024} (2305, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 3024,\ (\ :1/2),\ 0.580 - 0.814i)\)
\(L(1)\)  \(\approx\)  \(2.378400211\)
\(L(\frac12)\)  \(\approx\)  \(2.378400211\)
\(L(\frac{3}{2})\)   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 \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.5 - 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.5 - 7.79i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (5.5 + 9.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 16T + 79T^{2} \)
83 \( 1 + (-4.5 - 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.5 + 2.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-48.5 + 84.0i)T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.822878100594114865945204088360, −7.964870370083415265103848481142, −7.35597706933006908832816689482, −6.54054017613508464442419471293, −5.85705816305107194735936284920, −5.18065281311061335587419124054, −3.75368662891185918176227189437, −3.47690111054046174340751629999, −2.11695863038468567091529565257, −1.26717037014811580907718359026, 0.815088068354780966576537024946, 1.90845984032894267684025546185, 2.62298183179987522283068605653, 4.19204225946770728682520687994, 4.89398231997440069608512627800, 5.25443049034497508827519114832, 6.35021738247783955624437408215, 7.02064927995795923452348912758, 8.059120169174377305584619459297, 8.838851042366998806900759130497

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