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Properties

Label 2-105-35.9-c3-0-5
Degree $2$
Conductor $105$
Sign $-0.742 - 0.669i$
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  + (3.89 + 2.24i)2-s + (−2.59 + 1.5i)3-s + (6.10 + 10.5i)4-s + (−9.40 + 6.04i)5-s − 13.4·6-s + (3.71 + 18.1i)7-s + 18.9i·8-s + (4.5 − 7.79i)9-s + (−50.2 + 2.39i)10-s + (−11.2 − 19.4i)11-s + (−31.7 − 18.3i)12-s + 70.4i·13-s + (−26.3 + 78.9i)14-s + (15.3 − 29.8i)15-s + (6.30 − 10.9i)16-s + (52.3 − 30.2i)17-s + ⋯
L(s)  = 1  + (1.37 + 0.794i)2-s + (−0.499 + 0.288i)3-s + (0.763 + 1.32i)4-s + (−0.841 + 0.540i)5-s − 0.917·6-s + (0.200 + 0.979i)7-s + 0.836i·8-s + (0.166 − 0.288i)9-s + (−1.58 + 0.0757i)10-s + (−0.307 − 0.532i)11-s + (−0.763 − 0.440i)12-s + 1.50i·13-s + (−0.502 + 1.50i)14-s + (0.264 − 0.513i)15-s + (0.0985 − 0.170i)16-s + (0.747 − 0.431i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.742 - 0.669i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.742 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.742 - 0.669i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ -0.742 - 0.669i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.837112 + 2.17781i\)
\(L(\frac12)\) \(\approx\) \(0.837112 + 2.17781i\)
\(L(\frac{5}{2})\) 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 \( 1 + (9.40 - 6.04i)T \)
7 \( 1 + (-3.71 - 18.1i)T \)
good2 \( 1 + (-3.89 - 2.24i)T + (4 + 6.92i)T^{2} \)
11 \( 1 + (11.2 + 19.4i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 70.4iT - 2.19e3T^{2} \)
17 \( 1 + (-52.3 + 30.2i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (18.2 - 31.5i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-169. - 98.0i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 144.T + 2.43e4T^{2} \)
31 \( 1 + (88.3 + 153. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (190. + 109. i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 157.T + 6.89e4T^{2} \)
43 \( 1 - 99.6iT - 7.95e4T^{2} \)
47 \( 1 + (-372. - 214. i)T + (5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (360. - 208. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (385. + 668. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (46.2 - 80.0i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (23.7 - 13.6i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 388.T + 3.57e5T^{2} \)
73 \( 1 + (200. - 115. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (58.7 - 101. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 1.07e3iT - 5.71e5T^{2} \)
89 \( 1 + (93.5 - 162. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 415. iT - 9.12e5T^{2} \)
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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

−14.00210091360146530041277608287, −12.59893734701304599954170976251, −11.81723949938995566224858797615, −11.06519953132039613904665800374, −9.241426075913938259945208001157, −7.70233822620120948725669457973, −6.61574824224882069414560976004, −5.55219421794710190008940562545, −4.44297681022393821785718697879, −3.14292898298669447708344409839, 0.977048314666677281662635932004, 3.19794616572133762900020852645, 4.53319256401445271268242729883, 5.33983555349271226296043617755, 7.05415121855048979757242268500, 8.253499553118966540100453095095, 10.46606073976196886941507852644, 10.90161942258059884130964440242, 12.26512920837981646850419990388, 12.65480851443125214796403316831

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