Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.705 + 0.709i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0859 − 0.148i)2-s + (2.40 − 1.79i)3-s + (1.98 − 3.43i)4-s + (−1.39 + 4.80i)5-s + (−0.473 − 0.204i)6-s + (6.99 + 0.246i)7-s − 1.37·8-s + (2.58 − 8.61i)9-s + (0.834 − 0.205i)10-s + (−10.0 − 5.79i)11-s + (−1.37 − 11.8i)12-s + 7.34i·13-s + (−0.564 − 1.06i)14-s + (5.25 + 14.0i)15-s + (−7.82 − 13.5i)16-s + (2.30 − 3.99i)17-s + ⋯
L(s)  = 1  + (−0.0429 − 0.0744i)2-s + (0.802 − 0.596i)3-s + (0.496 − 0.859i)4-s + (−0.278 + 0.960i)5-s + (−0.0789 − 0.0340i)6-s + (0.999 + 0.0351i)7-s − 0.171·8-s + (0.287 − 0.957i)9-s + (0.0834 − 0.0205i)10-s + (−0.913 − 0.527i)11-s + (−0.114 − 0.985i)12-s + 0.565i·13-s + (−0.0403 − 0.0759i)14-s + (0.350 + 0.936i)15-s + (−0.488 − 0.846i)16-s + (0.135 − 0.234i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.705 + 0.709i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.705 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.705 + 0.709i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (74, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ 0.705 + 0.709i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.67167 - 0.695154i\)
\(L(\frac12)\)  \(\approx\)  \(1.67167 - 0.695154i\)
\(L(2)\)   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 \( 1 + (-2.40 + 1.79i)T \)
5 \( 1 + (1.39 - 4.80i)T \)
7 \( 1 + (-6.99 - 0.246i)T \)
good2 \( 1 + (0.0859 + 0.148i)T + (-2 + 3.46i)T^{2} \)
11 \( 1 + (10.0 + 5.79i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 - 7.34iT - 169T^{2} \)
17 \( 1 + (-2.30 + 3.99i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-5.93 - 10.2i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-11.8 - 20.5i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 32.7iT - 841T^{2} \)
31 \( 1 + (23.4 - 40.6i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (41.2 - 23.8i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 70.7iT - 1.68e3T^{2} \)
43 \( 1 - 14.1iT - 1.84e3T^{2} \)
47 \( 1 + (-26.1 - 45.2i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-11.5 + 19.9i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-1.09 - 0.633i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (32.3 + 55.9i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (17.1 + 9.91i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 48.1iT - 5.04e3T^{2} \)
73 \( 1 + (-107. - 62.2i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (34.4 + 59.6i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 35.8T + 6.88e3T^{2} \)
89 \( 1 + (-110. + 63.8i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 26.9iT - 9.40e3T^{2} \)
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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

−13.92423017435880965655594891691, −12.21487055667220874274221382417, −11.17744696602178375191211602495, −10.39504669061943001856191130349, −8.974721755512621236442766291759, −7.67217458044073727315815305113, −6.87664514967934110550120457984, −5.39186070018505223893904015970, −3.19946917127177306905732990489, −1.74482848418552191905893574784, 2.37681481772798576958592926976, 4.07280470223876063745102275601, 5.17370817694875340592633406847, 7.56264006788435471735429215297, 8.090833247941794019199441401887, 9.043994240651076587776499990432, 10.48230576698566454118425789556, 11.57879662802355078133413383197, 12.74566971754591699020841843040, 13.48627406434553202750757501610

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