Properties

Label 2-105-21.17-c3-0-1
Degree $2$
Conductor $105$
Sign $-0.728 - 0.685i$
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  + (−2.51 + 1.45i)2-s + (−2.26 − 4.67i)3-s + (0.214 − 0.371i)4-s + (−2.5 − 4.33i)5-s + (12.4 + 8.47i)6-s + (17.9 − 4.37i)7-s − 21.9i·8-s + (−16.7 + 21.1i)9-s + (12.5 + 7.25i)10-s + (−28.8 − 16.6i)11-s + (−2.22 − 0.162i)12-s + 62.4i·13-s + (−38.8 + 37.1i)14-s + (−14.5 + 21.4i)15-s + (33.6 + 58.2i)16-s + (−34.7 + 60.1i)17-s + ⋯
L(s)  = 1  + (−0.888 + 0.513i)2-s + (−0.435 − 0.900i)3-s + (0.0268 − 0.0464i)4-s + (−0.223 − 0.387i)5-s + (0.849 + 0.576i)6-s + (0.971 − 0.236i)7-s − 0.971i·8-s + (−0.620 + 0.783i)9-s + (0.397 + 0.229i)10-s + (−0.790 − 0.456i)11-s + (−0.0534 − 0.00391i)12-s + 1.33i·13-s + (−0.742 + 0.708i)14-s + (−0.251 + 0.369i)15-s + (0.525 + 0.909i)16-s + (−0.495 + 0.858i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.728 - 0.685i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.728 - 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.728 - 0.685i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ -0.728 - 0.685i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0923104 + 0.232787i\)
\(L(\frac12)\) \(\approx\) \(0.0923104 + 0.232787i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.26 + 4.67i)T \)
5 \( 1 + (2.5 + 4.33i)T \)
7 \( 1 + (-17.9 + 4.37i)T \)
good2 \( 1 + (2.51 - 1.45i)T + (4 - 6.92i)T^{2} \)
11 \( 1 + (28.8 + 16.6i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 62.4iT - 2.19e3T^{2} \)
17 \( 1 + (34.7 - 60.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (71.6 - 41.3i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (49.8 - 28.7i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 114. iT - 2.43e4T^{2} \)
31 \( 1 + (187. + 108. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-136. - 236. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 36.2T + 6.89e4T^{2} \)
43 \( 1 - 185.T + 7.95e4T^{2} \)
47 \( 1 + (25.2 + 43.6i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (204. + 117. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (267. - 462. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (313. - 181. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-232. + 403. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 1.07e3iT - 3.57e5T^{2} \)
73 \( 1 + (771. + 445. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (636. + 1.10e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 245.T + 5.71e5T^{2} \)
89 \( 1 + (548. + 950. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.08e3iT - 9.12e5T^{2} \)
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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

−13.50872144153657832442243797167, −12.66691842689945498774649024766, −11.53375006677665951462640210092, −10.55513945790759100836162999892, −8.867510175892692953179094422995, −8.136003674962866659456674409348, −7.29511639772359692350442651989, −6.03729909113665177209151342967, −4.35760882315371278044048749847, −1.62024547505654674259490559091, 0.19757049420474136176786141101, 2.54956092641370461768825149640, 4.64112735210822511308892674033, 5.65391358549506619092284563205, 7.70072655431833262854830855026, 8.748854056794905739217315409744, 9.856255862004576037172876780286, 10.83357892030694735569377426400, 11.17038450403888344966047729611, 12.51740778531019121251290818506

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