Properties

Label 2-105-21.5-c3-0-29
Degree $2$
Conductor $105$
Sign $-0.459 + 0.887i$
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  + (2.43 + 1.40i)2-s + (−2.49 − 4.55i)3-s + (−0.0626 − 0.108i)4-s + (−2.5 + 4.33i)5-s + (0.319 − 14.5i)6-s + (−12.1 − 13.9i)7-s − 22.8i·8-s + (−14.5 + 22.7i)9-s + (−12.1 + 7.01i)10-s + (−24.0 + 13.9i)11-s + (−0.338 + 0.556i)12-s − 85.9i·13-s + (−10.0 − 50.9i)14-s + (25.9 + 0.569i)15-s + (31.4 − 54.5i)16-s + (18.2 + 31.6i)17-s + ⋯
L(s)  = 1  + (0.859 + 0.496i)2-s + (−0.480 − 0.876i)3-s + (−0.00783 − 0.0135i)4-s + (−0.223 + 0.387i)5-s + (0.0217 − 0.991i)6-s + (−0.657 − 0.753i)7-s − 1.00i·8-s + (−0.537 + 0.843i)9-s + (−0.384 + 0.221i)10-s + (−0.660 + 0.381i)11-s + (−0.00813 + 0.0133i)12-s − 1.83i·13-s + (−0.191 − 0.973i)14-s + (0.447 + 0.00980i)15-s + (0.492 − 0.852i)16-s + (0.260 + 0.451i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.459 + 0.887i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.459 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.459 + 0.887i$
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: \((2,\ 105,\ (\ :3/2),\ -0.459 + 0.887i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.634145 - 1.04253i\)
\(L(\frac12)\) \(\approx\) \(0.634145 - 1.04253i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.49 + 4.55i)T \)
5 \( 1 + (2.5 - 4.33i)T \)
7 \( 1 + (12.1 + 13.9i)T \)
good2 \( 1 + (-2.43 - 1.40i)T + (4 + 6.92i)T^{2} \)
11 \( 1 + (24.0 - 13.9i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 85.9iT - 2.19e3T^{2} \)
17 \( 1 + (-18.2 - 31.6i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (55.3 + 31.9i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-100. - 58.2i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 80.6iT - 2.43e4T^{2} \)
31 \( 1 + (-268. + 155. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (41.0 - 71.0i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 85.8T + 6.89e4T^{2} \)
43 \( 1 + 94.2T + 7.95e4T^{2} \)
47 \( 1 + (-199. + 345. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (121. - 70.2i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (443. + 768. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-98.2 - 56.7i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-130. - 225. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 390. iT - 3.57e5T^{2} \)
73 \( 1 + (183. - 105. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-529. + 916. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 1.00e3T + 5.71e5T^{2} \)
89 \( 1 + (490. - 850. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.69e3iT - 9.12e5T^{2} \)
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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

−13.10241613649376993184003987649, −12.42228596831685668122403925184, −10.79356809502569831218595873156, −10.07870564415266359607356523926, −7.980093518508733577303301750448, −7.03156683928883792843291283022, −6.06867296786011080345334296085, −4.94101952830881961367612794597, −3.18214307236009092166027942561, −0.54274738426863843686560425590, 2.81329539794593940251750383040, 4.18335302342029017566168857003, 5.11955206261205035578099227457, 6.38731432592007607465302571131, 8.508604642855989362368532475178, 9.344656311138076380837881774914, 10.74419812716321317030651277774, 11.81918805608140695192363554071, 12.30924629058009195805309844393, 13.51278463680938350139348591273

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