Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.924 + 0.380i$
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 + (2.99 − 0.152i)3-s − 3i·4-s + (−4.24 − 2.63i)5-s + (2.22 + 2.01i)6-s + (5.49 − 4.33i)7-s + (4.94 − 4.94i)8-s + (8.95 − 0.915i)9-s + (−1.13 − 4.86i)10-s + 13.9i·11-s + (−0.458 − 8.98i)12-s + (−14.6 + 14.6i)13-s + (6.95 + 0.822i)14-s + (−13.1 − 7.25i)15-s − 4.99·16-s + (−4.86 + 4.86i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.353i)2-s + (0.998 − 0.0509i)3-s − 0.750i·4-s + (−0.849 − 0.527i)5-s + (0.371 + 0.335i)6-s + (0.785 − 0.619i)7-s + (0.618 − 0.618i)8-s + (0.994 − 0.101i)9-s + (−0.113 − 0.486i)10-s + 1.26i·11-s + (−0.0381 − 0.749i)12-s + (−1.12 + 1.12i)13-s + (0.496 + 0.0587i)14-s + (−0.875 − 0.483i)15-s − 0.312·16-s + (−0.286 + 0.286i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 + 0.380i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.924 + 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.924 + 0.380i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (83, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ 0.924 + 0.380i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.97061 - 0.389920i\)
\(L(\frac12)\)  \(\approx\)  \(1.97061 - 0.389920i\)
\(L(2)\)   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 \( 1 + (-2.99 + 0.152i)T \)
5 \( 1 + (4.24 + 2.63i)T \)
7 \( 1 + (-5.49 + 4.33i)T \)
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

−13.75221828547468941201701684967, −12.63056121882209179088885939853, −11.45880095701655499297220480624, −10.00776851551340900382934717187, −9.167437060716921000559712070464, −7.60041656713380529052303689674, −7.13064961762790519533381411619, −4.84140737884705436601899433501, −4.20474695791560271182040790583, −1.69677607090898950991512393254, 2.66872122165220419197177896454, 3.53322753007468996011627988743, 5.05947204640018669131588297483, 7.41881687558146766971625036911, 7.985977646168934936396580982500, 8.978224846525337066784238950485, 10.66162536891460101250759603623, 11.64709003691966044520666340516, 12.49045358358155124085602146037, 13.66500191028968240216631460783

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