Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.0338 + 0.999i$
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 + (−2.70 + 1.29i)3-s − 3i·4-s + (−0.707 − 4.94i)5-s + (−2.82 − i)6-s − 7·7-s + (4.94 − 4.94i)8-s + (5.65 − 7i)9-s + (3.00 − 4.00i)10-s − 9.89i·11-s + (3.87 + 8.12i)12-s + (−8 + 8i)13-s + (−4.94 − 4.94i)14-s + (8.31 + 12.4i)15-s − 4.99·16-s + (18.3 − 18.3i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.353i)2-s + (−0.902 + 0.430i)3-s − 0.750i·4-s + (−0.141 − 0.989i)5-s + (−0.471 − 0.166i)6-s − 7-s + (0.618 − 0.618i)8-s + (0.628 − 0.777i)9-s + (0.300 − 0.400i)10-s − 0.899i·11-s + (0.323 + 0.676i)12-s + (−0.615 + 0.615i)13-s + (−0.353 − 0.353i)14-s + (0.554 + 0.832i)15-s − 0.312·16-s + (1.08 − 1.08i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0338 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0338 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.0338 + 0.999i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (83, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ 0.0338 + 0.999i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.642616 - 0.621203i\)
\(L(\frac12)\)  \(\approx\)  \(0.642616 - 0.621203i\)
\(L(2)\)   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 \( 1 + (2.70 - 1.29i)T \)
5 \( 1 + (0.707 + 4.94i)T \)
7 \( 1 + 7T \)
good2 \( 1 + (-0.707 - 0.707i)T + 4iT^{2} \)
11 \( 1 + 9.89iT - 121T^{2} \)
13 \( 1 + (8 - 8i)T - 169iT^{2} \)
17 \( 1 + (-18.3 + 18.3i)T - 289iT^{2} \)
19 \( 1 - 10T + 361T^{2} \)
23 \( 1 + (24.0 - 24.0i)T - 529iT^{2} \)
29 \( 1 + 4.24T + 841T^{2} \)
31 \( 1 + 14iT - 961T^{2} \)
37 \( 1 + (-30 + 30i)T - 1.36e3iT^{2} \)
41 \( 1 - 33.9T + 1.68e3T^{2} \)
43 \( 1 + (-36 - 36i)T + 1.84e3iT^{2} \)
47 \( 1 + (-4.24 + 4.24i)T - 2.20e3iT^{2} \)
53 \( 1 + (-70.7 + 70.7i)T - 2.80e3iT^{2} \)
59 \( 1 + 29.6iT - 3.48e3T^{2} \)
61 \( 1 - 14iT - 3.72e3T^{2} \)
67 \( 1 + (-32 + 32i)T - 4.48e3iT^{2} \)
71 \( 1 + 59.3iT - 5.04e3T^{2} \)
73 \( 1 + (39 - 39i)T - 5.32e3iT^{2} \)
79 \( 1 - 56iT - 6.24e3T^{2} \)
83 \( 1 + (-36.7 - 36.7i)T + 6.88e3iT^{2} \)
89 \( 1 + 19.7iT - 7.92e3T^{2} \)
97 \( 1 + (113 + 113i)T + 9.40e3iT^{2} \)
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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

−13.29970524933299111543226286459, −12.19642228953249016864819279611, −11.29919337102225902588284417623, −9.750950037376473135474802278919, −9.477284155170749861766325078059, −7.37615866519857267870394514146, −5.97730715584268499076089613183, −5.32538334065141875136156967179, −3.97895784356392469049813223415, −0.66972576015080695041322104618, 2.60037500277158521014176848205, 4.08777774079155002828922460730, 5.84663728465937851000850026781, 7.06919363656686761701684223736, 7.85338330636812864145274429668, 10.00270778980951953464650537503, 10.66558550443880804843210896778, 12.14964233571588820502955470737, 12.35051281684918874588999107036, 13.44596570965769043601246493387

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