Properties

Label 2-105-105.104-c3-0-5
Degree $2$
Conductor $105$
Sign $0.665 - 0.746i$
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 + (39.9 + 42.1i)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 + (0.687 + 0.726i)15-s + 0.474·16-s + 0.513i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.665 - 0.746i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.665 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.665 - 0.746i$
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: \((2,\ 105,\ (\ :3/2),\ 0.665 - 0.746i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.352496 + 0.157878i\)
\(L(\frac12)\) \(\approx\) \(0.352496 + 0.157878i\)
\(L(\frac{5}{2})\) 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

−12.91445107287530402632513525156, −12.65115995432758976902211908580, −11.21423048657072670631748947447, −10.28329482316299990350398425669, −9.176163439397752364570890890737, −7.65186138292585430871937213157, −7.06472416644740653645943059490, −4.91797702207601105219501454462, −4.30141695626916132494301435813, −1.00772805300473887176682111939, 0.38966373706320837742333075659, 3.51027701890262727122774609869, 4.97563016749339784142878001153, 6.14132771194936151180299431213, 7.75359013747784967408252628240, 8.842790116812901057228870684810, 9.953360034678868208676837280793, 11.08832922456709611785022603920, 11.86464951013237577779929703916, 12.91532346764309602378990533367

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