Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.212 + 0.977i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.949 − 1.64i)2-s + (−2.65 + 1.40i)3-s + (0.196 − 0.340i)4-s + (4.51 + 2.14i)5-s + (4.82 + 3.02i)6-s + (4.60 − 5.26i)7-s − 8.34·8-s + (5.04 − 7.45i)9-s + (−0.759 − 9.46i)10-s + (−8.35 − 4.82i)11-s + (−0.0423 + 1.17i)12-s − 16.9i·13-s + (−13.0 − 2.57i)14-s + (−14.9 + 0.660i)15-s + (7.13 + 12.3i)16-s + (12.1 − 21.1i)17-s + ⋯
L(s)  = 1  + (−0.474 − 0.822i)2-s + (−0.883 + 0.468i)3-s + (0.0490 − 0.0850i)4-s + (0.903 + 0.429i)5-s + (0.804 + 0.504i)6-s + (0.658 − 0.752i)7-s − 1.04·8-s + (0.560 − 0.827i)9-s + (−0.0759 − 0.946i)10-s + (−0.759 − 0.438i)11-s + (−0.00353 + 0.0981i)12-s − 1.30i·13-s + (−0.931 − 0.184i)14-s + (−0.999 + 0.0440i)15-s + (0.446 + 0.772i)16-s + (0.716 − 1.24i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.212 + 0.977i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.212 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.212 + 0.977i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (74, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ -0.212 + 0.977i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.581838 - 0.722274i\)
\(L(\frac12)\)  \(\approx\)  \(0.581838 - 0.722274i\)
\(L(2)\)   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 \( 1 + (2.65 - 1.40i)T \)
5 \( 1 + (-4.51 - 2.14i)T \)
7 \( 1 + (-4.60 + 5.26i)T \)
good2 \( 1 + (0.949 + 1.64i)T + (-2 + 3.46i)T^{2} \)
11 \( 1 + (8.35 + 4.82i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + 16.9iT - 169T^{2} \)
17 \( 1 + (-12.1 + 21.1i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-6.95 - 12.0i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-0.354 - 0.614i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 16.5iT - 841T^{2} \)
31 \( 1 + (-7.12 + 12.3i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-1.08 + 0.626i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 24.1iT - 1.68e3T^{2} \)
43 \( 1 - 57.6iT - 1.84e3T^{2} \)
47 \( 1 + (-16.0 - 27.8i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (8.67 - 15.0i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (75.8 + 43.8i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-52.2 - 90.4i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (86.1 + 49.7i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 50.7iT - 5.04e3T^{2} \)
73 \( 1 + (-81.5 - 47.0i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (3.71 + 6.43i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 69.6T + 6.88e3T^{2} \)
89 \( 1 + (-78.3 + 45.2i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 90.4iT - 9.40e3T^{2} \)
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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

−13.03127657293516947431383602056, −11.73777808946155529057554645918, −10.86477885617987862970313168335, −10.24822162539249666561464349281, −9.565133945139789922722913564313, −7.70986115414742405220173196974, −6.07610651827143474096994998158, −5.18382668348681626060761930987, −3.02602863730993853941597758306, −0.948914907403487821616535923039, 1.97728881988840027011974878174, 5.03103233399995265292139218946, 5.96324009444984287587291229749, 7.02983178865043003558138267563, 8.228555575400302036792194771177, 9.253381051632133109302830510428, 10.62819121840921634929061655709, 11.97407342480196223726217203068, 12.54900763045933829940020548976, 13.82118125595232408699785560061

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