Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.126 - 0.991i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.38 + 0.637i)2-s + (−1.67 − 0.448i)3-s + (1.79 − 1.03i)4-s + (2.91 − 4.06i)5-s + 4.26·6-s + (−4.25 + 5.55i)7-s + (3.35 − 3.35i)8-s + (2.59 + 1.50i)9-s + (−4.34 + 11.5i)10-s + (4.95 + 8.57i)11-s + (−3.46 + 0.928i)12-s + (−14.1 + 14.1i)13-s + (6.57 − 15.9i)14-s + (−6.69 + 5.49i)15-s + (−9.99 + 17.3i)16-s + (−4.22 + 15.7i)17-s + ⋯
L(s)  = 1  + (−1.19 + 0.318i)2-s + (−0.557 − 0.149i)3-s + (0.448 − 0.259i)4-s + (0.582 − 0.812i)5-s + 0.711·6-s + (−0.607 + 0.794i)7-s + (0.419 − 0.419i)8-s + (0.288 + 0.166i)9-s + (−0.434 + 1.15i)10-s + (0.450 + 0.779i)11-s + (−0.288 + 0.0774i)12-s + (−1.08 + 1.08i)13-s + (0.469 − 1.13i)14-s + (−0.446 + 0.366i)15-s + (−0.624 + 1.08i)16-s + (−0.248 + 0.926i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.126 - 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.126 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.126 - 0.991i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (37, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ -0.126 - 0.991i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.316355 + 0.359310i\)
\(L(\frac12)\)  \(\approx\)  \(0.316355 + 0.359310i\)
\(L(2)\)   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 \( 1 + (1.67 + 0.448i)T \)
5 \( 1 + (-2.91 + 4.06i)T \)
7 \( 1 + (4.25 - 5.55i)T \)
good2 \( 1 + (2.38 - 0.637i)T + (3.46 - 2i)T^{2} \)
11 \( 1 + (-4.95 - 8.57i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (14.1 - 14.1i)T - 169iT^{2} \)
17 \( 1 + (4.22 - 15.7i)T + (-250. - 144.5i)T^{2} \)
19 \( 1 + (-23.3 - 13.4i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-1.79 - 6.70i)T + (-458. + 264.5i)T^{2} \)
29 \( 1 + 18.5iT - 841T^{2} \)
31 \( 1 + (-19.6 - 33.9i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-42.8 + 11.4i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 55.1T + 1.68e3T^{2} \)
43 \( 1 + (36.0 - 36.0i)T - 1.84e3iT^{2} \)
47 \( 1 + (-24.5 + 6.56i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-15.4 - 4.14i)T + (2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (36.0 - 20.8i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (51.6 - 89.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-1.98 + 7.39i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 58.5T + 5.04e3T^{2} \)
73 \( 1 + (19.6 + 5.26i)T + (4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (47.2 + 27.2i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-40.8 + 40.8i)T - 6.88e3iT^{2} \)
89 \( 1 + (49.6 + 28.6i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-37.4 - 37.4i)T + 9.40e3iT^{2} \)
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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

−13.65420872499003997640922047908, −12.51711810808483420112203377545, −11.85442052310000070478960347820, −9.987443902213010116030385788618, −9.584762116937723380217223381291, −8.571570803480787303067205910029, −7.19886962149951881367687616014, −6.08634486429319317958040010786, −4.60039097348148437663501745421, −1.66057728409250901596483116870, 0.58311756214274039450575533874, 2.95156287617157009468717210425, 5.16796200531896808278081568091, 6.71569353917613474598864037327, 7.67758576966836749917304866562, 9.392135754392261591065777369657, 9.972974462533926867317340596222, 10.81866189626283162274654248702, 11.69166999564399107452796390658, 13.34959346675730582278167112009

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