Properties

Label 2-105-21.5-c3-0-2
Degree $2$
Conductor $105$
Sign $-0.316 - 0.948i$
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  + (0.815 + 0.470i)2-s + (−4.77 − 2.06i)3-s + (−3.55 − 6.16i)4-s + (−2.5 + 4.33i)5-s + (−2.92 − 3.92i)6-s + (13.9 + 12.1i)7-s − 14.2i·8-s + (18.5 + 19.6i)9-s + (−4.07 + 2.35i)10-s + (−55.6 + 32.1i)11-s + (4.27 + 36.7i)12-s + 67.4i·13-s + (5.70 + 16.4i)14-s + (20.8 − 15.5i)15-s + (−21.7 + 37.6i)16-s + (−25.5 − 44.2i)17-s + ⋯
L(s)  = 1  + (0.288 + 0.166i)2-s + (−0.918 − 0.396i)3-s + (−0.444 − 0.770i)4-s + (−0.223 + 0.387i)5-s + (−0.198 − 0.267i)6-s + (0.755 + 0.655i)7-s − 0.629i·8-s + (0.685 + 0.728i)9-s + (−0.128 + 0.0744i)10-s + (−1.52 + 0.880i)11-s + (0.102 + 0.883i)12-s + 1.43i·13-s + (0.108 + 0.314i)14-s + (0.358 − 0.266i)15-s + (−0.339 + 0.588i)16-s + (−0.364 − 0.631i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.316 - 0.948i$
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: \((2,\ 105,\ (\ :3/2),\ -0.316 - 0.948i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.379551 + 0.526802i\)
\(L(\frac12)\) \(\approx\) \(0.379551 + 0.526802i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.77 + 2.06i)T \)
5 \( 1 + (2.5 - 4.33i)T \)
7 \( 1 + (-13.9 - 12.1i)T \)
good2 \( 1 + (-0.815 - 0.470i)T + (4 + 6.92i)T^{2} \)
11 \( 1 + (55.6 - 32.1i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 67.4iT - 2.19e3T^{2} \)
17 \( 1 + (25.5 + 44.2i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-99.0 - 57.1i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (74.0 + 42.7i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 138. iT - 2.43e4T^{2} \)
31 \( 1 + (23.9 - 13.8i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-63.3 + 109. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 72.8T + 6.89e4T^{2} \)
43 \( 1 + 550.T + 7.95e4T^{2} \)
47 \( 1 + (86.4 - 149. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (151. - 87.3i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (159. + 275. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-4.36 - 2.52i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (96.8 + 167. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 22.7iT - 3.57e5T^{2} \)
73 \( 1 + (-354. + 204. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (311. - 540. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 1.13e3T + 5.71e5T^{2} \)
89 \( 1 + (169. - 293. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 53.6iT - 9.12e5T^{2} \)
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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

−13.66949589053181037442828853172, −12.47875526808030513671650207425, −11.55302451782615487109232944628, −10.55614980702665527450277375720, −9.506388844309457151903810705381, −7.81579359430052205936927726896, −6.67988647731394369608907688420, −5.36770268732584231774976397179, −4.65671243608418931256017151915, −1.87621895767157212754405347647, 0.37087004009194212501084627039, 3.36798785807186564562659986254, 4.76001531934013803123372308067, 5.54970593259280274672643939814, 7.63114977650250403261247584542, 8.324782051387340402957976185869, 10.04247793184060235636773876888, 11.03301241963723703896549862196, 11.83172006744595314464659234694, 13.07446151594435276073244012675

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