Properties

Label 2-105-105.104-c3-0-2
Degree $2$
Conductor $105$
Sign $0.392 - 0.919i$
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  − 5.11·2-s + (−1.51 − 4.96i)3-s + 18.1·4-s + (−10.9 + 2.25i)5-s + (7.75 + 25.4i)6-s + (−11.4 − 14.5i)7-s − 51.7·8-s + (−22.3 + 15.0i)9-s + (55.9 − 11.5i)10-s − 55.4i·11-s + (−27.5 − 90.0i)12-s − 20.9·13-s + (58.6 + 74.3i)14-s + (27.8 + 51.0i)15-s + 119.·16-s + 96.8i·17-s + ⋯
L(s)  = 1  − 1.80·2-s + (−0.292 − 0.956i)3-s + 2.26·4-s + (−0.979 + 0.201i)5-s + (0.527 + 1.72i)6-s + (−0.619 − 0.785i)7-s − 2.28·8-s + (−0.829 + 0.558i)9-s + (1.77 − 0.364i)10-s − 1.51i·11-s + (−0.661 − 2.16i)12-s − 0.447·13-s + (1.11 + 1.41i)14-s + (0.478 + 0.877i)15-s + 1.86·16-s + 1.38i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.392 - 0.919i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.392 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.392 - 0.919i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ 0.392 - 0.919i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.106155 + 0.0700968i\)
\(L(\frac12)\) \(\approx\) \(0.106155 + 0.0700968i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.51 + 4.96i)T \)
5 \( 1 + (10.9 - 2.25i)T \)
7 \( 1 + (11.4 + 14.5i)T \)
good2 \( 1 + 5.11T + 8T^{2} \)
11 \( 1 + 55.4iT - 1.33e3T^{2} \)
13 \( 1 + 20.9T + 2.19e3T^{2} \)
17 \( 1 - 96.8iT - 4.91e3T^{2} \)
19 \( 1 - 33.1iT - 6.85e3T^{2} \)
23 \( 1 - 111.T + 1.21e4T^{2} \)
29 \( 1 - 156. iT - 2.43e4T^{2} \)
31 \( 1 + 80.4iT - 2.97e4T^{2} \)
37 \( 1 - 180. iT - 5.06e4T^{2} \)
41 \( 1 - 36.5T + 6.89e4T^{2} \)
43 \( 1 + 52.5iT - 7.95e4T^{2} \)
47 \( 1 + 259. iT - 1.03e5T^{2} \)
53 \( 1 + 191.T + 1.48e5T^{2} \)
59 \( 1 + 705.T + 2.05e5T^{2} \)
61 \( 1 - 427. iT - 2.26e5T^{2} \)
67 \( 1 - 306. iT - 3.00e5T^{2} \)
71 \( 1 - 513. iT - 3.57e5T^{2} \)
73 \( 1 + 360.T + 3.89e5T^{2} \)
79 \( 1 - 85.9T + 4.93e5T^{2} \)
83 \( 1 + 886. iT - 5.71e5T^{2} \)
89 \( 1 - 1.41e3T + 7.04e5T^{2} \)
97 \( 1 - 1.05e3T + 9.12e5T^{2} \)
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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

−13.21567902087830544418200509535, −12.03932162124804679019356227822, −11.02466473445559608325925628678, −10.48784969144820368768919095858, −8.809778809249886293223757147533, −8.021567679386225265163771013442, −7.14306588882689716030208198743, −6.21910662858860628029814647942, −3.17914194557567711660595687992, −1.04964532691864813022589998090, 0.16029871988625952209124786746, 2.78148305362832791510697435837, 4.84478089410233863267452088107, 6.74583787204258581858661123122, 7.77849436607473850443087656836, 9.279491354747717608048772528496, 9.404974151228319463612612752600, 10.70764106356668477716713792790, 11.69608927810435389639342962167, 12.38567735222375637471658448841

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