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Properties

Label 2-105-35.4-c3-0-17
Degree $2$
Conductor $105$
Sign $-0.639 + 0.768i$
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  + (−1.50 + 0.866i)2-s + (−2.59 − 1.5i)3-s + (−2.49 + 4.32i)4-s + (10.6 − 3.36i)5-s + 5.19·6-s + (−14.9 + 10.9i)7-s − 22.5i·8-s + (4.5 + 7.79i)9-s + (−13.0 + 14.2i)10-s + (−16.1 + 28.0i)11-s + (12.9 − 7.49i)12-s − 71.5i·13-s + (12.8 − 29.3i)14-s + (−32.7 − 7.24i)15-s + (−0.474 − 0.821i)16-s + (−79.8 − 46.1i)17-s + ⋯
L(s)  = 1  + (−0.530 + 0.306i)2-s + (−0.499 − 0.288i)3-s + (−0.312 + 0.540i)4-s + (0.953 − 0.301i)5-s + 0.353·6-s + (−0.805 + 0.592i)7-s − 0.995i·8-s + (0.166 + 0.288i)9-s + (−0.413 + 0.451i)10-s + (−0.443 + 0.767i)11-s + (0.312 − 0.180i)12-s − 1.52i·13-s + (0.245 − 0.561i)14-s + (−0.563 − 0.124i)15-s + (−0.00740 − 0.0128i)16-s + (−1.13 − 0.657i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.639 + 0.768i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.639 + 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.639 + 0.768i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ -0.639 + 0.768i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0923588 - 0.196916i\)
\(L(\frac12)\) \(\approx\) \(0.0923588 - 0.196916i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.59 + 1.5i)T \)
5 \( 1 + (-10.6 + 3.36i)T \)
7 \( 1 + (14.9 - 10.9i)T \)
good2 \( 1 + (1.50 - 0.866i)T + (4 - 6.92i)T^{2} \)
11 \( 1 + (16.1 - 28.0i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 71.5iT - 2.19e3T^{2} \)
17 \( 1 + (79.8 + 46.1i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (69.3 + 120. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (104. - 60.5i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 4.48T + 2.43e4T^{2} \)
31 \( 1 + (-9.30 + 16.1i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (237. - 136. i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 168.T + 6.89e4T^{2} \)
43 \( 1 - 176. iT - 7.95e4T^{2} \)
47 \( 1 + (-151. + 87.6i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (213. + 123. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (381. - 661. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-69.6 - 120. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-133. - 77.2i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 610.T + 3.57e5T^{2} \)
73 \( 1 + (952. + 550. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (642. + 1.11e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 264. iT - 5.71e5T^{2} \)
89 \( 1 + (-366. - 635. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 772. iT - 9.12e5T^{2} \)
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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

−13.07195156665425969904151615237, −12.15180319892601792430645868931, −10.48367613397176964990350350438, −9.541924314207695018483070335971, −8.631400782008147294352992431369, −7.25687280479856735776885796554, −6.16416570361804132573193223729, −4.84222474653047241458749051159, −2.62899627021028212099943578124, −0.14524535750147078839043346444, 1.91345921906773891040919245782, 4.17199052728018971068914993465, 5.84814272250238133609868720598, 6.56405528625084791030440285629, 8.623479581203925193077770324780, 9.613393353367441189523094046649, 10.44830730224374997466247178965, 11.03155535291891645801654827314, 12.62386665416871438835888957089, 13.83940735916147642905174721628

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