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Properties

Label 2-105-35.9-c3-0-10
Degree $2$
Conductor $105$
Sign $0.962 - 0.269i$
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  + (1.96 + 1.13i)2-s + (−2.59 + 1.5i)3-s + (−1.43 − 2.48i)4-s + (11.1 + 0.0158i)5-s − 6.79·6-s + (18.1 − 3.88i)7-s − 24.6i·8-s + (4.5 − 7.79i)9-s + (21.9 + 12.6i)10-s + (18.9 + 32.8i)11-s + (7.46 + 4.30i)12-s + 45.0i·13-s + (39.9 + 12.8i)14-s + (−29.0 + 16.7i)15-s + (16.3 − 28.3i)16-s + (108. − 62.8i)17-s + ⋯
L(s)  = 1  + (0.693 + 0.400i)2-s + (−0.499 + 0.288i)3-s + (−0.179 − 0.310i)4-s + (0.999 + 0.00141i)5-s − 0.462·6-s + (0.977 − 0.209i)7-s − 1.08i·8-s + (0.166 − 0.288i)9-s + (0.692 + 0.401i)10-s + (0.519 + 0.900i)11-s + (0.179 + 0.103i)12-s + 0.962i·13-s + (0.761 + 0.246i)14-s + (−0.500 + 0.287i)15-s + (0.256 − 0.443i)16-s + (1.55 − 0.897i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.269i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.962 - 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.962 - 0.269i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ 0.962 - 0.269i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.27686 + 0.312910i\)
\(L(\frac12)\) \(\approx\) \(2.27686 + 0.312910i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.59 - 1.5i)T \)
5 \( 1 + (-11.1 - 0.0158i)T \)
7 \( 1 + (-18.1 + 3.88i)T \)
good2 \( 1 + (-1.96 - 1.13i)T + (4 + 6.92i)T^{2} \)
11 \( 1 + (-18.9 - 32.8i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 45.0iT - 2.19e3T^{2} \)
17 \( 1 + (-108. + 62.8i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (8.92 - 15.4i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (69.7 + 40.2i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 206.T + 2.43e4T^{2} \)
31 \( 1 + (140. + 243. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-144. - 83.1i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 188.T + 6.89e4T^{2} \)
43 \( 1 - 339. iT - 7.95e4T^{2} \)
47 \( 1 + (70.2 + 40.5i)T + (5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (496. - 286. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-114. - 198. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-110. + 191. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (198. - 114. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 933.T + 3.57e5T^{2} \)
73 \( 1 + (-646. + 373. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (386. - 669. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 801. iT - 5.71e5T^{2} \)
89 \( 1 + (-219. + 380. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 380. iT - 9.12e5T^{2} \)
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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

−13.59521809384740939849908687633, −12.40858045121467145724404589343, −11.30326930253279575383922029157, −9.930149660814881639248012584102, −9.417446235886918079968229684147, −7.40124642763576070957256216224, −6.16004629177239830676398690900, −5.18667768977671545301849536808, −4.21903630221471594718440703979, −1.53231468298208757619205801491, 1.65945294200323362874746178563, 3.45817961213463119853541681495, 5.24624365434918265770925381083, 5.83389619603166599220589322126, 7.75487397069123028278923524225, 8.785304109235513390651852931848, 10.38778725798929230706549272577, 11.32963610952257579451232865163, 12.33581404942591304345793250324, 13.11681455709582277988045832468

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