Properties

Label 2-105-21.5-c3-0-11
Degree $2$
Conductor $105$
Sign $-0.930 + 0.366i$
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  + (−4.56 − 2.63i)2-s + (−4.97 − 1.51i)3-s + (9.88 + 17.1i)4-s + (−2.5 + 4.33i)5-s + (18.7 + 19.9i)6-s + (12.9 − 13.2i)7-s − 62.0i·8-s + (22.4 + 15.0i)9-s + (22.8 − 13.1i)10-s + (4.67 − 2.70i)11-s + (−23.2 − 100. i)12-s − 11.1i·13-s + (−93.9 + 26.3i)14-s + (18.9 − 17.7i)15-s + (−84.2 + 146. i)16-s + (49.0 + 84.9i)17-s + ⋯
L(s)  = 1  + (−1.61 − 0.931i)2-s + (−0.956 − 0.291i)3-s + (1.23 + 2.13i)4-s + (−0.223 + 0.387i)5-s + (1.27 + 1.36i)6-s + (0.698 − 0.715i)7-s − 2.74i·8-s + (0.830 + 0.556i)9-s + (0.721 − 0.416i)10-s + (0.128 − 0.0740i)11-s + (−0.559 − 2.40i)12-s − 0.237i·13-s + (−1.79 + 0.503i)14-s + (0.326 − 0.305i)15-s + (−1.31 + 2.28i)16-s + (0.699 + 1.21i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.930 + 0.366i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.930 + 0.366i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.930 + 0.366i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ -0.930 + 0.366i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0651375 - 0.343155i\)
\(L(\frac12)\) \(\approx\) \(0.0651375 - 0.343155i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.97 + 1.51i)T \)
5 \( 1 + (2.5 - 4.33i)T \)
7 \( 1 + (-12.9 + 13.2i)T \)
good2 \( 1 + (4.56 + 2.63i)T + (4 + 6.92i)T^{2} \)
11 \( 1 + (-4.67 + 2.70i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 11.1iT - 2.19e3T^{2} \)
17 \( 1 + (-49.0 - 84.9i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (38.3 + 22.1i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (161. + 93.0i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 210. iT - 2.43e4T^{2} \)
31 \( 1 + (-143. + 82.8i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-37.5 + 65.0i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 149.T + 6.89e4T^{2} \)
43 \( 1 + 415.T + 7.95e4T^{2} \)
47 \( 1 + (-276. + 478. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-116. + 67.5i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-284. - 493. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (403. + 233. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (379. + 656. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 113. iT - 3.57e5T^{2} \)
73 \( 1 + (590. - 341. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-649. + 1.12e3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 143.T + 5.71e5T^{2} \)
89 \( 1 + (-103. + 178. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.04e3iT - 9.12e5T^{2} \)
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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

−12.12891425782731323409607922793, −11.58015745302471394869696527974, −10.39607352355182238175308063144, −10.23963540698398179907181906871, −8.317057536165110959957642477655, −7.63974400887264718564735524033, −6.36582413335112315071998106755, −4.03911545681481562664119994531, −1.90591663338165695657818422054, −0.40611757919927463547602211737, 1.36968744059234123718743727715, 4.98548476379110073951307711069, 5.97104160955413123603480396780, 7.21731704458045099772554228205, 8.308987392718921204541635434551, 9.361531905999370409305919772129, 10.24232290131835318488378954875, 11.43142192399647990247961580169, 12.11909407269211059945445220375, 14.24072738058302443549099067525

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