Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.270 - 0.962i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.860 − 0.496i)2-s + (−2.67 − 1.35i)3-s + (−1.50 + 2.60i)4-s + (−1.93 + 1.11i)5-s + (−2.97 + 0.162i)6-s + (−1.66 + 6.79i)7-s + 6.96i·8-s + (5.32 + 7.25i)9-s + (−1.11 + 1.92i)10-s + (0.568 + 0.328i)11-s + (7.57 − 4.94i)12-s − 10.1·13-s + (1.94 + 6.67i)14-s + (6.69 − 0.366i)15-s + (−2.56 − 4.44i)16-s + (−16.8 − 9.72i)17-s + ⋯
L(s)  = 1  + (0.430 − 0.248i)2-s + (−0.892 − 0.451i)3-s + (−0.376 + 0.652i)4-s + (−0.387 + 0.223i)5-s + (−0.495 + 0.0271i)6-s + (−0.238 + 0.971i)7-s + 0.870i·8-s + (0.591 + 0.806i)9-s + (−0.111 + 0.192i)10-s + (0.0517 + 0.0298i)11-s + (0.630 − 0.411i)12-s − 0.781·13-s + (0.138 + 0.476i)14-s + (0.446 − 0.0244i)15-s + (−0.160 − 0.278i)16-s + (−0.990 − 0.571i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.270 - 0.962i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.270 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.270 - 0.962i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ -0.270 - 0.962i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.424123 + 0.559927i\)
\(L(\frac12)\)  \(\approx\)  \(0.424123 + 0.559927i\)
\(L(2)\)   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 \( 1 + (2.67 + 1.35i)T \)
5 \( 1 + (1.93 - 1.11i)T \)
7 \( 1 + (1.66 - 6.79i)T \)
good2 \( 1 + (-0.860 + 0.496i)T + (2 - 3.46i)T^{2} \)
11 \( 1 + (-0.568 - 0.328i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + 10.1T + 169T^{2} \)
17 \( 1 + (16.8 + 9.72i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-9.11 - 15.7i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (3.29 - 1.90i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 - 50.8iT - 841T^{2} \)
31 \( 1 + (-26.8 + 46.5i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-7.81 - 13.5i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 57.3iT - 1.68e3T^{2} \)
43 \( 1 - 65.7T + 1.84e3T^{2} \)
47 \( 1 + (-22.4 + 12.9i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (5.64 + 3.25i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-18.7 - 10.8i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-17.1 - 29.6i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (39.5 - 68.5i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 39.0iT - 5.04e3T^{2} \)
73 \( 1 + (19.9 - 34.5i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-31.9 - 55.2i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 75.6iT - 6.88e3T^{2} \)
89 \( 1 + (57.0 - 32.9i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 113.T + 9.40e3T^{2} \)
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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

−13.50280579261830397972343821108, −12.54810719911263887184112098576, −11.94329916764036754208655210468, −11.15942433328023029712352387467, −9.581194908422188788369330405568, −8.208224453264407804043155204582, −7.07665812416725183487447416262, −5.63970035693800081951709863277, −4.46268920625270719685893664059, −2.63053995235147838026611905269, 0.50973459560718989310457399594, 4.06731986467854327599309851849, 4.83881658794397855876871050279, 6.20604236435395593561288325314, 7.24037229005889885816311618110, 9.156576613435316294459058418440, 10.17066827183494818854292768490, 10.97503955855803787702275752819, 12.24307278422834410377314670679, 13.26682172082213554381422167938

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