Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.756 - 0.654i$
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 + (1.29 + 2.70i)3-s + 3i·4-s + (−0.707 + 4.94i)5-s + (−2.82 − i)6-s − 7i·7-s + (−4.94 − 4.94i)8-s + (−5.65 + 7i)9-s + (−3.00 − 4.00i)10-s − 9.89i·11-s + (−8.12 + 3.87i)12-s + (8 + 8i)13-s + (4.94 + 4.94i)14-s + (−14.3 + 4.48i)15-s − 4.99·16-s + (18.3 + 18.3i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.353i)2-s + (0.430 + 0.902i)3-s + 0.750i·4-s + (−0.141 + 0.989i)5-s + (−0.471 − 0.166i)6-s i·7-s + (−0.618 − 0.618i)8-s + (−0.628 + 0.777i)9-s + (−0.300 − 0.400i)10-s − 0.899i·11-s + (−0.676 + 0.323i)12-s + (0.615 + 0.615i)13-s + (0.353 + 0.353i)14-s + (−0.954 + 0.299i)15-s − 0.312·16-s + (1.08 + 1.08i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.756 - 0.654i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.756 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.756 - 0.654i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (62, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ -0.756 - 0.654i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.406414 + 1.09037i\)
\(L(\frac12)\)  \(\approx\)  \(0.406414 + 1.09037i\)
\(L(2)\)   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 \( 1 + (-1.29 - 2.70i)T \)
5 \( 1 + (0.707 - 4.94i)T \)
7 \( 1 + 7iT \)
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

−14.07678509670913698784115097051, −13.17493378571447988228762422563, −11.43258225213499663048712549907, −10.72502973846190124713599141829, −9.597015855697787738483434200141, −8.367731524258278712629340586843, −7.51340445053640932574810495743, −6.18924078400127978636880539657, −3.98132795514445681081630975342, −3.24997244599020296648130090940, 1.00988996578530781446821528905, 2.55924115136985596871101929537, 5.06184352957247083015334092420, 6.18178216206250038301597745392, 7.82499744274678226631490166019, 8.908446731241995236782187308874, 9.543373303150697688468989652902, 11.14164475424363069891522428439, 12.31858788737258624574846345451, 12.76639049567333604254548808238

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