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Properties

Label 2-105-21.17-c3-0-19
Degree $2$
Conductor $105$
Sign $0.817 + 0.576i$
Analytic cond. $6.19520$
Root an. cond. $2.48901$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.815 + 0.470i)2-s + (−0.600 − 5.16i)3-s + (−3.55 + 6.16i)4-s + (2.5 + 4.33i)5-s + (2.92 + 3.92i)6-s + (13.9 − 12.1i)7-s − 14.2i·8-s + (−26.2 + 6.20i)9-s + (−4.07 − 2.35i)10-s + (55.6 + 32.1i)11-s + (33.9 + 14.6i)12-s − 67.4i·13-s + (−5.70 + 16.4i)14-s + (20.8 − 15.5i)15-s + (−21.7 − 37.6i)16-s + (25.5 − 44.2i)17-s + ⋯
L(s)  = 1  + (−0.288 + 0.166i)2-s + (−0.115 − 0.993i)3-s + (−0.444 + 0.770i)4-s + (0.223 + 0.387i)5-s + (0.198 + 0.267i)6-s + (0.755 − 0.655i)7-s − 0.629i·8-s + (−0.973 + 0.229i)9-s + (−0.128 − 0.0744i)10-s + (1.52 + 0.880i)11-s + (0.816 + 0.352i)12-s − 1.43i·13-s + (−0.108 + 0.314i)14-s + (0.358 − 0.266i)15-s + (−0.339 − 0.588i)16-s + (0.364 − 0.631i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.817 + 0.576i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.817 + 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.817 + 0.576i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ 0.817 + 0.576i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.28052 - 0.405918i\)
\(L(\frac12)\) \(\approx\) \(1.28052 - 0.405918i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.600 + 5.16i)T \)
5 \( 1 + (-2.5 - 4.33i)T \)
7 \( 1 + (-13.9 + 12.1i)T \)
good2 \( 1 + (0.815 - 0.470i)T + (4 - 6.92i)T^{2} \)
11 \( 1 + (-55.6 - 32.1i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 67.4iT - 2.19e3T^{2} \)
17 \( 1 + (-25.5 + 44.2i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-99.0 + 57.1i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-74.0 + 42.7i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 138. iT - 2.43e4T^{2} \)
31 \( 1 + (23.9 + 13.8i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-63.3 - 109. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 72.8T + 6.89e4T^{2} \)
43 \( 1 + 550.T + 7.95e4T^{2} \)
47 \( 1 + (-86.4 - 149. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-151. - 87.3i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-159. + 275. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-4.36 + 2.52i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (96.8 - 167. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 22.7iT - 3.57e5T^{2} \)
73 \( 1 + (-354. - 204. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (311. + 540. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 1.13e3T + 5.71e5T^{2} \)
89 \( 1 + (-169. - 293. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 53.6iT - 9.12e5T^{2} \)
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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

−13.19663770514777777787839081475, −12.18242416828494885157428762646, −11.32312789301436379958442258296, −9.819096826201552890379619891351, −8.571261969508435656039875616216, −7.44560465394972211322034053973, −6.89218829945892984555537794910, −5.00957547611814005025753754155, −3.20209280133599639704784013676, −1.05158022514648804491399629090, 1.46285958182162979801816857685, 3.95528994837995441302500998585, 5.20478992084934757771564041200, 6.14293325064804579197666964308, 8.537367909426952504978814738706, 9.137695901993237943392683700531, 9.951565977286481444727101826037, 11.40578996477689006553923258514, 11.73995051655872893361482559946, 13.86678881893359103628308556728

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