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Properties

Label 2-105-105.104-c3-0-27
Degree $2$
Conductor $105$
Sign $-0.0728 + 0.997i$
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  + 1.22·2-s + (−4.91 + 1.67i)3-s − 6.50·4-s + (9.89 + 5.21i)5-s + (−6.01 + 2.04i)6-s + (−1.55 − 18.4i)7-s − 17.7·8-s + (21.4 − 16.4i)9-s + (12.0 + 6.37i)10-s − 54.5i·11-s + (32.0 − 10.8i)12-s − 24.4·13-s + (−1.90 − 22.5i)14-s + (−57.3 − 9.09i)15-s + 30.3·16-s − 36.0i·17-s + ⋯
L(s)  = 1  + 0.432·2-s + (−0.946 + 0.322i)3-s − 0.813·4-s + (0.884 + 0.466i)5-s + (−0.409 + 0.139i)6-s + (−0.0841 − 0.996i)7-s − 0.783·8-s + (0.792 − 0.609i)9-s + (0.382 + 0.201i)10-s − 1.49i·11-s + (0.769 − 0.261i)12-s − 0.522·13-s + (−0.0363 − 0.430i)14-s + (−0.987 − 0.156i)15-s + 0.474·16-s − 0.513i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0728 + 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0728 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.0728 + 0.997i$
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: \((2,\ 105,\ (\ :3/2),\ -0.0728 + 0.997i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.607585 - 0.653558i\)
\(L(\frac12)\) \(\approx\) \(0.607585 - 0.653558i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.91 - 1.67i)T \)
5 \( 1 + (-9.89 - 5.21i)T \)
7 \( 1 + (1.55 + 18.4i)T \)
good2 \( 1 - 1.22T + 8T^{2} \)
11 \( 1 + 54.5iT - 1.33e3T^{2} \)
13 \( 1 + 24.4T + 2.19e3T^{2} \)
17 \( 1 + 36.0iT - 4.91e3T^{2} \)
19 \( 1 + 99.5iT - 6.85e3T^{2} \)
23 \( 1 + 97.3T + 1.21e4T^{2} \)
29 \( 1 + 14.1iT - 2.43e4T^{2} \)
31 \( 1 - 186. iT - 2.97e4T^{2} \)
37 \( 1 + 199. iT - 5.06e4T^{2} \)
41 \( 1 + 313.T + 6.89e4T^{2} \)
43 \( 1 - 30.0iT - 7.95e4T^{2} \)
47 \( 1 + 514. iT - 1.03e5T^{2} \)
53 \( 1 - 451.T + 1.48e5T^{2} \)
59 \( 1 + 355.T + 2.05e5T^{2} \)
61 \( 1 - 656. iT - 2.26e5T^{2} \)
67 \( 1 - 73.7iT - 3.00e5T^{2} \)
71 \( 1 + 611. iT - 3.57e5T^{2} \)
73 \( 1 + 424.T + 3.89e5T^{2} \)
79 \( 1 - 814.T + 4.93e5T^{2} \)
83 \( 1 + 585. iT - 5.71e5T^{2} \)
89 \( 1 - 732.T + 7.04e5T^{2} \)
97 \( 1 - 272.T + 9.12e5T^{2} \)
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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

−13.38136387842914802467297791941, −11.95416915871345978842509069563, −10.78945185996494567999068131898, −10.01206213011122021107025774007, −8.914405970659879888465468722682, −7.00576035140359838995599978101, −5.87153765773719271517987999871, −4.86901400778772563480556885226, −3.44151091645560458361559380250, −0.49749988105666134184196493778, 1.90796872178350465262848159755, 4.48813761260439556300813140551, 5.43885298497457419444341364039, 6.29755975071203480033634607950, 8.069512192374756372162264368695, 9.542046022647849384770189353002, 10.10512973678741701298914806988, 12.07037589541300946061728005710, 12.43411384603477498749080201682, 13.26260502023974140073843355418

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