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Properties

Label 2-105-105.83-c2-0-8
Degree $2$
Conductor $105$
Sign $0.338 + 0.940i$
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  + (−1.88 − 1.88i)2-s + (−2.65 + 1.39i)3-s + 3.12i·4-s + (4.96 − 0.551i)5-s + (7.64 + 2.36i)6-s + (2.09 + 6.68i)7-s + (−1.65 + 1.65i)8-s + (5.08 − 7.42i)9-s + (−10.4 − 8.33i)10-s − 17.9i·11-s + (−4.36 − 8.28i)12-s + (11.1 − 11.1i)13-s + (8.66 − 16.5i)14-s + (−12.4 + 8.41i)15-s + 18.7·16-s + (−0.666 + 0.666i)17-s + ⋯
L(s)  = 1  + (−0.943 − 0.943i)2-s + (−0.884 + 0.466i)3-s + 0.780i·4-s + (0.993 − 0.110i)5-s + (1.27 + 0.394i)6-s + (0.298 + 0.954i)7-s + (−0.207 + 0.207i)8-s + (0.564 − 0.825i)9-s + (−1.04 − 0.833i)10-s − 1.62i·11-s + (−0.363 − 0.690i)12-s + (0.856 − 0.856i)13-s + (0.618 − 1.18i)14-s + (−0.827 + 0.561i)15-s + 1.17·16-s + (−0.0392 + 0.0392i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.338 + 0.940i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.338 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.338 + 0.940i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ 0.338 + 0.940i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.624261 - 0.438809i\)
\(L(\frac12)\) \(\approx\) \(0.624261 - 0.438809i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.65 - 1.39i)T \)
5 \( 1 + (-4.96 + 0.551i)T \)
7 \( 1 + (-2.09 - 6.68i)T \)
good2 \( 1 + (1.88 + 1.88i)T + 4iT^{2} \)
11 \( 1 + 17.9iT - 121T^{2} \)
13 \( 1 + (-11.1 + 11.1i)T - 169iT^{2} \)
17 \( 1 + (0.666 - 0.666i)T - 289iT^{2} \)
19 \( 1 - 10.8T + 361T^{2} \)
23 \( 1 + (-2.20 + 2.20i)T - 529iT^{2} \)
29 \( 1 - 22.9T + 841T^{2} \)
31 \( 1 - 26.1iT - 961T^{2} \)
37 \( 1 + (-41.6 + 41.6i)T - 1.36e3iT^{2} \)
41 \( 1 - 6.85T + 1.68e3T^{2} \)
43 \( 1 + (37.6 + 37.6i)T + 1.84e3iT^{2} \)
47 \( 1 + (-5.55 + 5.55i)T - 2.20e3iT^{2} \)
53 \( 1 + (32.4 - 32.4i)T - 2.80e3iT^{2} \)
59 \( 1 - 99.8iT - 3.48e3T^{2} \)
61 \( 1 + 44.6iT - 3.72e3T^{2} \)
67 \( 1 + (18.0 - 18.0i)T - 4.48e3iT^{2} \)
71 \( 1 - 6.35iT - 5.04e3T^{2} \)
73 \( 1 + (55.3 - 55.3i)T - 5.32e3iT^{2} \)
79 \( 1 - 59.7iT - 6.24e3T^{2} \)
83 \( 1 + (42.2 + 42.2i)T + 6.88e3iT^{2} \)
89 \( 1 + 58.4iT - 7.92e3T^{2} \)
97 \( 1 + (11.1 + 11.1i)T + 9.40e3iT^{2} \)
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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

−12.94852398519582045016239765734, −11.84274163158206044847465246109, −10.98521811138716901436969075656, −10.32170712886967032307673813426, −9.164076211318021226514080767820, −8.479594320619737664948198692038, −6.02674631261738232519205059910, −5.47224879849863647328698754379, −3.01933909615240945527606931228, −1.05239878435453956070183159159, 1.42634916627372273614260621228, 4.70784869879552249997403515865, 6.28529964783701949352716738106, 6.93457674666200202602755615306, 7.923064735339431079222233681567, 9.537855213709129800914199048143, 10.18673175979384183916953131901, 11.46923496350068406221271930582, 12.80557008008652069630423653222, 13.72725080487804235769402282870

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