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Properties

Label 2-105-21.17-c3-0-25
Degree $2$
Conductor $105$
Sign $0.404 + 0.914i$
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  + (3.06 − 1.76i)2-s + (4.26 − 2.96i)3-s + (2.25 − 3.91i)4-s + (2.5 + 4.33i)5-s + (7.83 − 16.6i)6-s + (5.77 − 17.5i)7-s + 12.3i·8-s + (9.42 − 25.3i)9-s + (15.3 + 8.84i)10-s + (−3.05 − 1.76i)11-s + (−1.95 − 23.3i)12-s − 14.5i·13-s + (−13.4 − 64.1i)14-s + (23.5 + 11.0i)15-s + (39.8 + 69.0i)16-s + (−30.9 + 53.5i)17-s + ⋯
L(s)  = 1  + (1.08 − 0.625i)2-s + (0.821 − 0.570i)3-s + (0.282 − 0.488i)4-s + (0.223 + 0.387i)5-s + (0.532 − 1.13i)6-s + (0.311 − 0.950i)7-s + 0.544i·8-s + (0.349 − 0.937i)9-s + (0.484 + 0.279i)10-s + (−0.0837 − 0.0483i)11-s + (−0.0470 − 0.562i)12-s − 0.310i·13-s + (−0.256 − 1.22i)14-s + (0.404 + 0.190i)15-s + (0.622 + 1.07i)16-s + (−0.441 + 0.764i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.404 + 0.914i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.404 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.404 + 0.914i$
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: \((2,\ 105,\ (\ :3/2),\ 0.404 + 0.914i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.88141 - 1.87585i\)
\(L(\frac12)\) \(\approx\) \(2.88141 - 1.87585i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.26 + 2.96i)T \)
5 \( 1 + (-2.5 - 4.33i)T \)
7 \( 1 + (-5.77 + 17.5i)T \)
good2 \( 1 + (-3.06 + 1.76i)T + (4 - 6.92i)T^{2} \)
11 \( 1 + (3.05 + 1.76i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 14.5iT - 2.19e3T^{2} \)
17 \( 1 + (30.9 - 53.5i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (39.8 - 23.0i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (99.5 - 57.4i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 127. iT - 2.43e4T^{2} \)
31 \( 1 + (-183. - 106. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (142. + 246. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 328.T + 6.89e4T^{2} \)
43 \( 1 + 108.T + 7.95e4T^{2} \)
47 \( 1 + (-194. - 337. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (514. + 297. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (275. - 477. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-411. + 237. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-208. + 360. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 399. iT - 3.57e5T^{2} \)
73 \( 1 + (134. + 77.6i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (586. + 1.01e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 4.43T + 5.71e5T^{2} \)
89 \( 1 + (505. + 875. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 27.5iT - 9.12e5T^{2} \)
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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

−13.11587784447864721591150154774, −12.44842052875021615192325684328, −11.15185957437274732119509484627, −10.19228208605948712328151773213, −8.551540955293928893260763722327, −7.50776604568335844397202104332, −6.11352404310041436821918086201, −4.33400659781999160731494804397, −3.26739124036056775205810431589, −1.81061611041807287204383425055, 2.50015275393949225592063709550, 4.23759523133141316033744508150, 5.10592020687088936715769518814, 6.38101094062171129549077105275, 7.972387375993439473988904004909, 9.075861356950345872070273152320, 10.03211401820359586111062645943, 11.68471538136352716554265313966, 12.82925799114489105761251070857, 13.75532413534331383585157489002

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