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Properties

Label 2-105-35.24-c2-0-8
Degree $2$
Conductor $105$
Sign $0.866 + 0.499i$
Analytic cond. $2.86104$
Root an. cond. $1.69146$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.08 + 1.20i)2-s + (0.866 − 1.5i)3-s + (0.910 − 1.57i)4-s + (4.48 − 2.20i)5-s + 4.17i·6-s + (−6.98 − 0.420i)7-s − 5.25i·8-s + (−1.5 − 2.59i)9-s + (−6.70 + 10.0i)10-s + (6.92 − 11.9i)11-s + (−1.57 − 2.73i)12-s + 20.5·13-s + (15.1 − 7.55i)14-s + (0.570 − 8.64i)15-s + (9.98 + 17.2i)16-s + (6.03 − 10.4i)17-s + ⋯
L(s)  = 1  + (−1.04 + 0.603i)2-s + (0.288 − 0.5i)3-s + (0.227 − 0.394i)4-s + (0.897 − 0.441i)5-s + 0.696i·6-s + (−0.998 − 0.0600i)7-s − 0.657i·8-s + (−0.166 − 0.288i)9-s + (−0.670 + 1.00i)10-s + (0.629 − 1.08i)11-s + (−0.131 − 0.227i)12-s + 1.58·13-s + (1.07 − 0.539i)14-s + (0.0380 − 0.576i)15-s + (0.623 + 1.08i)16-s + (0.355 − 0.615i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.499i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.866 + 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.866 + 0.499i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (94, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ 0.866 + 0.499i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.872222 - 0.233353i\)
\(L(\frac12)\) \(\approx\) \(0.872222 - 0.233353i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 + 1.5i)T \)
5 \( 1 + (-4.48 + 2.20i)T \)
7 \( 1 + (6.98 + 0.420i)T \)
good2 \( 1 + (2.08 - 1.20i)T + (2 - 3.46i)T^{2} \)
11 \( 1 + (-6.92 + 11.9i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 20.5T + 169T^{2} \)
17 \( 1 + (-6.03 + 10.4i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (10.7 - 6.21i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-16.5 + 9.56i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 1.74T + 841T^{2} \)
31 \( 1 + (42.2 + 24.3i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (20.4 - 11.8i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 70.2iT - 1.68e3T^{2} \)
43 \( 1 - 45.3iT - 1.84e3T^{2} \)
47 \( 1 + (-8.98 - 15.5i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (57.9 + 33.4i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (30.7 + 17.7i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-51.9 + 30.0i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-57.3 - 33.1i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 31.6T + 5.04e3T^{2} \)
73 \( 1 + (9.99 - 17.3i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-49.4 - 85.6i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 92.2T + 6.88e3T^{2} \)
89 \( 1 + (-67.5 + 39.0i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 93.8T + 9.40e3T^{2} \)
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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

−13.27635396489020099889268832536, −12.82603751455452314468796343797, −11.08393966243284838976530378828, −9.702560623616623461001885222435, −8.990573042524157356024622578228, −8.200187356894512076876148975680, −6.61975919305659493626885897470, −6.04329290200089018630344967423, −3.46651236993658296645573834978, −1.02465341161685420851670982424, 1.84406465670577499336318631508, 3.51937373066544074146244019241, 5.64182783567181505933812366450, 6.98329406166399208301280021435, 8.844383982927695772186122872191, 9.299195956399607060205015325344, 10.35992480043767375875034667776, 10.88785252179997818304363938639, 12.47542015197271686871934502761, 13.62064563230150461759418620667

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