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Properties

Label 2-105-35.12-c1-0-4
Degree $2$
Conductor $105$
Sign $-0.698 + 0.715i$
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  + (−2.24 + 0.602i)2-s + (0.258 − 0.965i)3-s + (2.95 − 1.70i)4-s + (−2.22 − 0.198i)5-s + 2.32i·6-s + (−2.59 + 0.519i)7-s + (−2.33 + 2.33i)8-s + (−0.866 − 0.499i)9-s + (5.12 − 0.895i)10-s + (−1.76 − 3.05i)11-s + (−0.884 − 3.30i)12-s + (−4.49 − 4.49i)13-s + (5.51 − 2.73i)14-s + (−0.767 + 2.10i)15-s + (0.421 − 0.729i)16-s + (1.79 + 0.481i)17-s + ⋯
L(s)  = 1  + (−1.58 + 0.425i)2-s + (0.149 − 0.557i)3-s + (1.47 − 0.854i)4-s + (−0.996 − 0.0886i)5-s + 0.950i·6-s + (−0.980 + 0.196i)7-s + (−0.824 + 0.824i)8-s + (−0.288 − 0.166i)9-s + (1.62 − 0.283i)10-s + (−0.531 − 0.921i)11-s + (−0.255 − 0.952i)12-s + (−1.24 − 1.24i)13-s + (1.47 − 0.730i)14-s + (−0.198 + 0.542i)15-s + (0.105 − 0.182i)16-s + (0.436 + 0.116i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.698 + 0.715i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.698 + 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.698 + 0.715i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ -0.698 + 0.715i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0714110 - 0.169466i\)
\(L(\frac12)\) \(\approx\) \(0.0714110 - 0.169466i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 + (2.22 + 0.198i)T \)
7 \( 1 + (2.59 - 0.519i)T \)
good2 \( 1 + (2.24 - 0.602i)T + (1.73 - i)T^{2} \)
11 \( 1 + (1.76 + 3.05i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.49 + 4.49i)T + 13iT^{2} \)
17 \( 1 + (-1.79 - 0.481i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.0699 + 0.121i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.997 - 3.72i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 2.01iT - 29T^{2} \)
31 \( 1 + (4.56 - 2.63i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.61 + 1.50i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 0.903iT - 41T^{2} \)
43 \( 1 + (-2.38 + 2.38i)T - 43iT^{2} \)
47 \( 1 + (0.639 + 2.38i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.71 - 0.726i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (3.15 + 5.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (8.69 + 5.01i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.77 - 10.3i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 5.09T + 71T^{2} \)
73 \( 1 + (-2.42 + 9.04i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (7.30 + 4.21i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.37 + 7.37i)T + 83iT^{2} \)
89 \( 1 + (1.75 - 3.03i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.70 + 8.70i)T - 97iT^{2} \)
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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

−13.17244088343750125291028308694, −12.24072710708675577859264333004, −10.98150746002923504146387021235, −9.944596837377226844616150482421, −8.830919090854090623857190300254, −7.82694767806291808792857413537, −7.23853691126031514017747445073, −5.76778108904111806428023651723, −3.06011566819787825771212054859, −0.32213996771203634690653618302, 2.70603372100374099910490999155, 4.45493966206129708834201288853, 6.97529184431999035506710427166, 7.71721106593196489785001355769, 9.087012603807190114769498748658, 9.777566114202905782190973981793, 10.65323571164457967674783332692, 11.76579180452958673484348989550, 12.62639758520591588516388659776, 14.47825282625861734031925308667

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