Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{2} \cdot 5 $
Sign $-0.101 + 0.994i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.11 − 0.726i)5-s + 4.05i·7-s + 0.985i·11-s − 4.94·13-s − 4.52i·17-s − 2.60i·19-s − 3.53i·23-s + (3.94 + 3.07i)25-s − 7.59i·29-s + 3.28·31-s + (2.94 − 8.58i)35-s + 0.945·37-s − 0.568·41-s + 8.45·43-s + 2.60i·47-s + ⋯
L(s)  = 1  + (−0.945 − 0.324i)5-s + 1.53i·7-s + 0.297i·11-s − 1.37·13-s − 1.09i·17-s − 0.597i·19-s − 0.737i·23-s + (0.789 + 0.614i)25-s − 1.41i·29-s + 0.589·31-s + (0.497 − 1.45i)35-s + 0.155·37-s − 0.0887·41-s + 1.29·43-s + 0.379i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.101 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.101 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
\( \varepsilon \)  =  $-0.101 + 0.994i$
motivic weight  =  \(1\)
character  :  $\chi_{1440} (1009, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1440,\ (\ :1/2),\ -0.101 + 0.994i)\)
\(L(1)\)  \(\approx\)  \(0.6752674629\)
\(L(\frac12)\)  \(\approx\)  \(0.6752674629\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (2.11 + 0.726i)T \)
good7 \( 1 - 4.05iT - 7T^{2} \)
11 \( 1 - 0.985iT - 11T^{2} \)
13 \( 1 + 4.94T + 13T^{2} \)
17 \( 1 + 4.52iT - 17T^{2} \)
19 \( 1 + 2.60iT - 19T^{2} \)
23 \( 1 + 3.53iT - 23T^{2} \)
29 \( 1 + 7.59iT - 29T^{2} \)
31 \( 1 - 3.28T + 31T^{2} \)
37 \( 1 - 0.945T + 37T^{2} \)
41 \( 1 + 0.568T + 41T^{2} \)
43 \( 1 - 8.45T + 43T^{2} \)
47 \( 1 - 2.60iT - 47T^{2} \)
53 \( 1 - 0.229T + 53T^{2} \)
59 \( 1 + 9.10iT - 59T^{2} \)
61 \( 1 + 11.0iT - 61T^{2} \)
67 \( 1 + 8.45T + 67T^{2} \)
71 \( 1 - 1.43T + 71T^{2} \)
73 \( 1 + 11.9iT - 73T^{2} \)
79 \( 1 + 3.28T + 79T^{2} \)
83 \( 1 + 9.89T + 83T^{2} \)
89 \( 1 + 12.3T + 89T^{2} \)
97 \( 1 + 3.23iT - 97T^{2} \)
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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

−9.317422231550736392106797964458, −8.525019473411241508308538362903, −7.75857054893916644466120625468, −7.01629729079947900323297345952, −5.95515433413025904233717980270, −4.93971405099376057097786525951, −4.48742785455651352512099117769, −2.95466635946635248793854288529, −2.30279813199603973688039819981, −0.29588614584358726466541293078, 1.20762817364506137780143491830, 2.88391517053527830766255718598, 3.88653999397147373953762516926, 4.39829816710960920915233857087, 5.58481999941133922192195739680, 6.80423973831906134361232788116, 7.35686775293356819598562405348, 7.912814185676388373336194658450, 8.835540875988732251772129975086, 10.03404218383304999373678784662

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