Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.254 + 0.967i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 0.707i)2-s + (−0.152 + 2.99i)3-s − 3i·4-s + (−4.24 − 2.63i)5-s + (2.22 − 2.01i)6-s + (4.33 − 5.49i)7-s + (−4.94 + 4.94i)8-s + (−8.95 − 0.915i)9-s + (1.13 + 4.86i)10-s − 13.9i·11-s + (8.98 + 0.458i)12-s + (14.6 − 14.6i)13-s + (−6.95 + 0.822i)14-s + (8.55 − 12.3i)15-s − 4.99·16-s + (−4.86 + 4.86i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.353i)2-s + (−0.0509 + 0.998i)3-s − 0.750i·4-s + (−0.849 − 0.527i)5-s + (0.371 − 0.335i)6-s + (0.619 − 0.785i)7-s + (−0.618 + 0.618i)8-s + (−0.994 − 0.101i)9-s + (0.113 + 0.486i)10-s − 1.26i·11-s + (0.749 + 0.0381i)12-s + (1.12 − 1.12i)13-s + (−0.496 + 0.0587i)14-s + (0.570 − 0.821i)15-s − 0.312·16-s + (−0.286 + 0.286i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.254 + 0.967i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.254 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.254 + 0.967i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (83, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ -0.254 + 0.967i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.492599 - 0.638850i\)
\(L(\frac12)\)  \(\approx\)  \(0.492599 - 0.638850i\)
\(L(2)\)   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 \( 1 + (0.152 - 2.99i)T \)
5 \( 1 + (4.24 + 2.63i)T \)
7 \( 1 + (-4.33 + 5.49i)T \)
good2 \( 1 + (0.707 + 0.707i)T + 4iT^{2} \)
11 \( 1 + 13.9iT - 121T^{2} \)
13 \( 1 + (-14.6 + 14.6i)T - 169iT^{2} \)
17 \( 1 + (4.86 - 4.86i)T - 289iT^{2} \)
19 \( 1 + 21.7T + 361T^{2} \)
23 \( 1 + (1.77 - 1.77i)T - 529iT^{2} \)
29 \( 1 - 28.0T + 841T^{2} \)
31 \( 1 - 17.2iT - 961T^{2} \)
37 \( 1 + (6.50 - 6.50i)T - 1.36e3iT^{2} \)
41 \( 1 + 26.7T + 1.68e3T^{2} \)
43 \( 1 + (-33.1 - 33.1i)T + 1.84e3iT^{2} \)
47 \( 1 + (-18.5 + 18.5i)T - 2.20e3iT^{2} \)
53 \( 1 + (-48.3 + 48.3i)T - 2.80e3iT^{2} \)
59 \( 1 + 29.6iT - 3.48e3T^{2} \)
61 \( 1 - 21.0iT - 3.72e3T^{2} \)
67 \( 1 + (32.4 - 32.4i)T - 4.48e3iT^{2} \)
71 \( 1 + 16.0iT - 5.04e3T^{2} \)
73 \( 1 + (-57.3 + 57.3i)T - 5.32e3iT^{2} \)
79 \( 1 + 75.8iT - 6.24e3T^{2} \)
83 \( 1 + (-51.9 - 51.9i)T + 6.88e3iT^{2} \)
89 \( 1 + 174. iT - 7.92e3T^{2} \)
97 \( 1 + (16.6 + 16.6i)T + 9.40e3iT^{2} \)
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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

−13.34319271497122349213222750142, −11.65052288502401838284224639242, −10.83101485191203102685721216455, −10.41802395397084741362698157420, −8.718159598571372894813343841126, −8.313813022354988603622733823404, −6.06354016854790123068234386086, −4.83160063783118934216517813793, −3.53438050103299432174730216890, −0.69441233491994569917528715843, 2.34730008857012980853675029065, 4.22477815554135633892250999252, 6.40315074342101693263290385781, 7.20566728872118239609741023652, 8.237041892639130579162612370061, 8.958614996450881933714277800428, 11.02685788620364851239012983390, 11.95856616758321110568912022591, 12.46878125553842883238638634864, 13.79555767013381708788236909876

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