Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.999 - 0.0272i$
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 + (−0.346 + 2.97i)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 + (−8.75 − 2.06i)9-s + (0.834 − 0.205i)10-s + (10.0 + 5.79i)11-s + (9.55 + 7.10i)12-s + 7.34i·13-s + (0.564 + 1.06i)14-s + (13.8 + 5.81i)15-s + (−7.82 − 13.5i)16-s + (−2.30 + 3.99i)17-s + ⋯
L(s)  = 1  + (0.0429 + 0.0744i)2-s + (−0.115 + 0.993i)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 + (−0.973 − 0.229i)9-s + (0.0834 − 0.0205i)10-s + (0.913 + 0.527i)11-s + (0.796 + 0.592i)12-s + 0.565i·13-s + (0.0403 + 0.0759i)14-s + (0.921 + 0.387i)15-s + (−0.488 − 0.846i)16-s + (−0.135 + 0.234i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0272i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0272i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.999 - 0.0272i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (74, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ 0.999 - 0.0272i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.62090 + 0.0220553i\)
\(L(\frac12)\)  \(\approx\)  \(1.62090 + 0.0220553i\)
\(L(2)\)   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 \( 1 + (0.346 - 2.97i)T \)
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

−13.98877306580544699124862503417, −12.13984212096117049355438580187, −11.40787082352389953964144400053, −10.23580344860069664454010228709, −9.388086830644242377993240929890, −8.311630693284546638625862904302, −6.42440467249729205251894156770, −5.19882446279882635159121995110, −4.34257427551471719250403925881, −1.67338324119277151891412897142, 1.97234751824795294200281441552, 3.44308569350822566897823314318, 5.72869856640379854518908638578, 7.01830304412441268820072709316, 7.67292324809671421860471302009, 8.893706891093634533336527927189, 10.89648275288475245274942157853, 11.41908961341896396626031454920, 12.33805253677113899243698212874, 13.59560642372022547587270749055

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