Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.795 - 0.606i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.31 − 1.91i)2-s + (1.84 − 2.36i)3-s + (5.34 + 9.24i)4-s + (−1.93 − 1.11i)5-s + (−10.6 + 4.32i)6-s + (−5.86 − 3.82i)7-s − 25.5i·8-s + (−2.21 − 8.72i)9-s + (4.28 + 7.41i)10-s + (−9.48 + 5.47i)11-s + (31.7 + 4.38i)12-s − 3.75·13-s + (12.1 + 23.9i)14-s + (−6.21 + 2.52i)15-s + (−27.6 + 47.9i)16-s + (−12.6 + 7.30i)17-s + ⋯
L(s)  = 1  + (−1.65 − 0.957i)2-s + (0.613 − 0.789i)3-s + (1.33 + 2.31i)4-s + (−0.387 − 0.223i)5-s + (−1.77 + 0.721i)6-s + (−0.837 − 0.546i)7-s − 3.19i·8-s + (−0.246 − 0.969i)9-s + (0.428 + 0.741i)10-s + (−0.862 + 0.497i)11-s + (2.64 + 0.365i)12-s − 0.288·13-s + (0.866 + 1.70i)14-s + (−0.414 + 0.168i)15-s + (−1.72 + 2.99i)16-s + (−0.744 + 0.429i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.795 - 0.606i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.795 - 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.795 - 0.606i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (86, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ -0.795 - 0.606i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.102353 + 0.302932i\)
\(L(\frac12)\)  \(\approx\)  \(0.102353 + 0.302932i\)
\(L(2)\)   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 \( 1 + (-1.84 + 2.36i)T \)
5 \( 1 + (1.93 + 1.11i)T \)
7 \( 1 + (5.86 + 3.82i)T \)
good2 \( 1 + (3.31 + 1.91i)T + (2 + 3.46i)T^{2} \)
11 \( 1 + (9.48 - 5.47i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 3.75T + 169T^{2} \)
17 \( 1 + (12.6 - 7.30i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-5.54 + 9.60i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-8.53 - 4.92i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 10.0iT - 841T^{2} \)
31 \( 1 + (12.0 + 20.9i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-19.1 + 33.2i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 67.5iT - 1.68e3T^{2} \)
43 \( 1 + 77.4T + 1.84e3T^{2} \)
47 \( 1 + (4.91 + 2.83i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-59.9 + 34.6i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-45.4 + 26.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-6.77 + 11.7i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (10.6 + 18.4i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 25.6iT - 5.04e3T^{2} \)
73 \( 1 + (-12.4 - 21.4i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-55.3 + 95.8i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 102. iT - 6.88e3T^{2} \)
89 \( 1 + (-62.5 - 36.1i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 6.23T + 9.40e3T^{2} \)
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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

−12.76437480010416430362488763439, −11.71938570131529088238365406595, −10.56572390529543919880464166908, −9.547670934100648339711485501576, −8.664391056497726013382731692983, −7.57564407591114422199938398690, −6.90730661030544996604547756291, −3.57510178609152622542542671561, −2.23299704710115587268685220481, −0.35006913497195498604336474585, 2.74802015293590296969405803742, 5.27044746161433015155934498076, 6.64822383882793017143666443261, 7.909391868158892201375951147092, 8.700456188942691134336268106781, 9.665942347736498689599168919634, 10.39999432153316909499280289057, 11.47517170399654942875165809953, 13.48270762050486707519615843340, 14.85878359298249112315743063233

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