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Properties

Label 2-105-35.3-c1-0-0
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: $\chi_{105} (73, \cdot )$
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

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

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