Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.963 + 0.266i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.334 − 0.192i)2-s + (1.42 + 0.983i)3-s + (−0.925 − 1.60i)4-s + (0.5 − 0.866i)5-s + (−0.286 − 0.603i)6-s + (2.36 − 1.17i)7-s + 1.48i·8-s + (1.06 + 2.80i)9-s + (−0.334 + 0.192i)10-s + (−2.20 + 1.27i)11-s + (0.257 − 3.19i)12-s + 3.06i·13-s + (−1.01 − 0.0640i)14-s + (1.56 − 0.742i)15-s + (−1.56 + 2.71i)16-s + (−3.23 − 5.59i)17-s + ⋯
L(s)  = 1  + (−0.236 − 0.136i)2-s + (0.823 + 0.567i)3-s + (−0.462 − 0.801i)4-s + (0.223 − 0.387i)5-s + (−0.116 − 0.246i)6-s + (0.895 − 0.444i)7-s + 0.525i·8-s + (0.354 + 0.934i)9-s + (−0.105 + 0.0609i)10-s + (−0.663 + 0.383i)11-s + (0.0743 − 0.922i)12-s + 0.850i·13-s + (−0.272 − 0.0171i)14-s + (0.404 − 0.191i)15-s + (−0.391 + 0.677i)16-s + (−0.783 − 1.35i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 + 0.266i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.963 + 0.266i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.963 + 0.266i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (26, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ 0.963 + 0.266i)$
$L(1)$  $\approx$  $1.09836 - 0.149135i$
$L(\frac12)$  $\approx$  $1.09836 - 0.149135i$
$L(\frac{3}{2})$   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 \( 1 + (-1.42 - 0.983i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-2.36 + 1.17i)T \)
good2 \( 1 + (0.334 + 0.192i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (2.20 - 1.27i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.06iT - 13T^{2} \)
17 \( 1 + (3.23 + 5.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.03 + 0.597i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.64 + 1.52i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.77iT - 29T^{2} \)
31 \( 1 + (5.95 - 3.43i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.77 + 3.07i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.31T + 41T^{2} \)
43 \( 1 - 5.46T + 43T^{2} \)
47 \( 1 + (1.61 - 2.78i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-11.4 + 6.62i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.98 + 3.43i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (8.08 + 4.67i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.75 - 3.04i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.921iT - 71T^{2} \)
73 \( 1 + (-0.256 + 0.148i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.14 - 7.18i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 2.11T + 83T^{2} \)
89 \( 1 + (-9.41 + 16.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 12.3iT - 97T^{2} \)
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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

−14.07874107555876725204632861580, −13.00206837158709626919272116331, −11.23756137423981301994665667537, −10.41092456395354983101861467668, −9.323443785780582253204961223840, −8.650685888680432109751747939164, −7.27913426347532731119326670792, −5.16637667998238812178339920386, −4.41423813479039682933611043540, −2.03334558200098910459550346423, 2.43595711252732681082797155052, 3.99637367614305659200120332665, 5.97281425600697849800202054507, 7.64952366460940468959301626423, 8.166208195851317724220497558656, 9.140118084312437840244033116311, 10.55694582075965613072632514729, 11.95353043287279428414690975188, 13.01920173943693079996171539715, 13.61454001675492953099948478475

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