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Properties

Label 2-105-7.3-c2-0-1
Degree $2$
Conductor $105$
Sign $0.997 + 0.0643i$
Analytic cond. $2.86104$
Root an. cond. $1.69146$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.99 − 3.44i)2-s + (1.5 + 0.866i)3-s + (−5.92 + 10.2i)4-s + (−1.93 + 1.11i)5-s − 6.89i·6-s + (2.28 + 6.61i)7-s + 31.2·8-s + (1.5 + 2.59i)9-s + (7.70 + 4.45i)10-s + (−1.73 + 2.99i)11-s + (−17.7 + 10.2i)12-s + 11.3i·13-s + (18.2 − 21.0i)14-s − 3.87·15-s + (−38.5 − 66.7i)16-s + (4.14 + 2.39i)17-s + ⋯
L(s)  = 1  + (−0.995 − 1.72i)2-s + (0.5 + 0.288i)3-s + (−1.48 + 2.56i)4-s + (−0.387 + 0.223i)5-s − 1.14i·6-s + (0.326 + 0.945i)7-s + 3.90·8-s + (0.166 + 0.288i)9-s + (0.770 + 0.445i)10-s + (−0.157 + 0.272i)11-s + (−1.48 + 0.855i)12-s + 0.876i·13-s + (1.30 − 1.50i)14-s − 0.258·15-s + (−2.40 − 4.17i)16-s + (0.243 + 0.140i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0643i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0643i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.997 + 0.0643i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ 0.997 + 0.0643i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.769574 - 0.0247886i\)
\(L(\frac12)\) \(\approx\) \(0.769574 - 0.0247886i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.5 - 0.866i)T \)
5 \( 1 + (1.93 - 1.11i)T \)
7 \( 1 + (-2.28 - 6.61i)T \)
good2 \( 1 + (1.99 + 3.44i)T + (-2 + 3.46i)T^{2} \)
11 \( 1 + (1.73 - 2.99i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 11.3iT - 169T^{2} \)
17 \( 1 + (-4.14 - 2.39i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (1.06 - 0.613i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-7.08 - 12.2i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 29.3T + 841T^{2} \)
31 \( 1 + (-34.9 - 20.1i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-12.4 - 21.6i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 12.0iT - 1.68e3T^{2} \)
43 \( 1 - 4.34T + 1.84e3T^{2} \)
47 \( 1 + (-46.6 + 26.9i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-12.8 + 22.3i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (81.6 + 47.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-44.0 + 25.4i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-1.22 + 2.12i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 96.9T + 5.04e3T^{2} \)
73 \( 1 + (6.88 + 3.97i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (52.3 + 90.6i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 57.4iT - 6.88e3T^{2} \)
89 \( 1 + (-43.6 + 25.2i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 50.7iT - 9.40e3T^{2} \)
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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

−13.10732618868597626229109565673, −12.04192395985851760625117315427, −11.40685484386345403028204287628, −10.30275298123208932187025393783, −9.294628070985948482363756603703, −8.556178734238594283560360864394, −7.50020372446689643125754469662, −4.60511420496719658448527586978, −3.24999911480929526357292058401, −1.94926393498238356457365248524, 0.810386405034463025555121546550, 4.42468759799854666239149545681, 5.86533221807429559275772265032, 7.25389609489954589989891212232, 7.85046322848228703150203143222, 8.750921139877547213995318025504, 9.933141929383760879058702905170, 10.91343409484990354697701693470, 13.07065281621689297678369525491, 13.91840935395956853300417915840

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