Properties

Label 2-105-21.17-c1-0-3
Degree $2$
Conductor $105$
Sign $0.963 - 0.266i$
Analytic cond. $0.838429$
Root an. cond. $0.915657$
Motivic weight $1$
Arithmetic yes
Rational no
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

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.963 - 0.266i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ 0.963 - 0.266i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09836 + 0.149135i\)
\(L(\frac12)\) \(\approx\) \(1.09836 + 0.149135i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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