Properties

Degree 2
Conductor $ 3^{2} \cdot 5 \cdot 7 $
Sign $0.953 - 0.301i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.540 − 0.540i)2-s − 1.41i·4-s + (−1.03 + 1.98i)5-s + (0.614 + 2.57i)7-s + (−1.84 + 1.84i)8-s + (1.63 − 0.510i)10-s + 3.85·11-s + (3.66 + 3.66i)13-s + (1.05 − 1.72i)14-s − 0.839·16-s + (1.49 − 1.49i)17-s − 0.0697·19-s + (2.80 + 1.46i)20-s + (−2.08 − 2.08i)22-s + (0.534 − 0.534i)23-s + ⋯
L(s)  = 1  + (−0.381 − 0.381i)2-s − 0.708i·4-s + (−0.463 + 0.886i)5-s + (0.232 + 0.972i)7-s + (−0.652 + 0.652i)8-s + (0.515 − 0.161i)10-s + 1.16·11-s + (1.01 + 1.01i)13-s + (0.282 − 0.460i)14-s − 0.209·16-s + (0.361 − 0.361i)17-s − 0.0160·19-s + (0.627 + 0.328i)20-s + (−0.443 − 0.443i)22-s + (0.111 − 0.111i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 - 0.301i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.953 - 0.301i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.953 - 0.301i$
motivic weight  =  \(1\)
character  :  $\chi_{315} (307, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 315,\ (\ :1/2),\ 0.953 - 0.301i)\)
\(L(1)\)  \(\approx\)  \(1.02789 + 0.158413i\)
\(L(\frac12)\)  \(\approx\)  \(1.02789 + 0.158413i\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (1.03 - 1.98i)T \)
7 \( 1 + (-0.614 - 2.57i)T \)
good2 \( 1 + (0.540 + 0.540i)T + 2iT^{2} \)
11 \( 1 - 3.85T + 11T^{2} \)
13 \( 1 + (-3.66 - 3.66i)T + 13iT^{2} \)
17 \( 1 + (-1.49 + 1.49i)T - 17iT^{2} \)
19 \( 1 + 0.0697T + 19T^{2} \)
23 \( 1 + (-0.534 + 0.534i)T - 23iT^{2} \)
29 \( 1 + 2.77iT - 29T^{2} \)
31 \( 1 - 2.39iT - 31T^{2} \)
37 \( 1 + (-6.18 - 6.18i)T + 37iT^{2} \)
41 \( 1 - 8.68iT - 41T^{2} \)
43 \( 1 + (2.77 - 2.77i)T - 43iT^{2} \)
47 \( 1 + (-5.49 + 5.49i)T - 47iT^{2} \)
53 \( 1 + (6.13 - 6.13i)T - 53iT^{2} \)
59 \( 1 + 6.97T + 59T^{2} \)
61 \( 1 + 14.3iT - 61T^{2} \)
67 \( 1 + (-0.416 - 0.416i)T + 67iT^{2} \)
71 \( 1 - 8.12T + 71T^{2} \)
73 \( 1 + (9.55 + 9.55i)T + 73iT^{2} \)
79 \( 1 + 9.86iT - 79T^{2} \)
83 \( 1 + (-1.63 - 1.63i)T + 83iT^{2} \)
89 \( 1 + 5.05T + 89T^{2} \)
97 \( 1 + (6.85 - 6.85i)T - 97iT^{2} \)
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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

−11.50568916732456431644931100058, −11.00296817831610248652395238892, −9.762166220503648793163151624376, −9.092080464215874605347581846919, −8.130856810763385518080771367620, −6.58678767876453050936256615049, −6.07995396386984031079813200317, −4.54091141449912247870904097168, −3.06133799526270148835223533165, −1.62889064105508872080527662447, 0.988496640476410646023729276727, 3.57099995595241382622245453935, 4.20122629324645072988156042571, 5.80191886933433919176431768726, 7.06115833056456167206426434192, 7.890416586920725533904004821888, 8.615285226309457363952399417083, 9.449438193871837928096010483218, 10.74887105081933519895606726878, 11.66059989662360258074479494320

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