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Properties

Label 2-105-105.62-c2-0-3
Degree $2$
Conductor $105$
Sign $-0.254 - 0.967i$
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  + (−0.707 + 0.707i)2-s + (−0.152 − 2.99i)3-s + 3i·4-s + (−4.24 + 2.63i)5-s + (2.22 + 2.01i)6-s + (4.33 + 5.49i)7-s + (−4.94 − 4.94i)8-s + (−8.95 + 0.915i)9-s + (1.13 − 4.86i)10-s + 13.9i·11-s + (8.98 − 0.458i)12-s + (14.6 + 14.6i)13-s + (−6.95 − 0.822i)14-s + (8.55 + 12.3i)15-s − 4.99·16-s + (−4.86 − 4.86i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.353i)2-s + (−0.0509 − 0.998i)3-s + 0.750i·4-s + (−0.849 + 0.527i)5-s + (0.371 + 0.335i)6-s + (0.619 + 0.785i)7-s + (−0.618 − 0.618i)8-s + (−0.994 + 0.101i)9-s + (0.113 − 0.486i)10-s + 1.26i·11-s + (0.749 − 0.0381i)12-s + (1.12 + 1.12i)13-s + (−0.496 − 0.0587i)14-s + (0.570 + 0.821i)15-s − 0.312·16-s + (−0.286 − 0.286i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.254 - 0.967i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.254 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.254 - 0.967i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ -0.254 - 0.967i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.492599 + 0.638850i\)
\(L(\frac12)\) \(\approx\) \(0.492599 + 0.638850i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.152 + 2.99i)T \)
5 \( 1 + (4.24 - 2.63i)T \)
7 \( 1 + (-4.33 - 5.49i)T \)
good2 \( 1 + (0.707 - 0.707i)T - 4iT^{2} \)
11 \( 1 - 13.9iT - 121T^{2} \)
13 \( 1 + (-14.6 - 14.6i)T + 169iT^{2} \)
17 \( 1 + (4.86 + 4.86i)T + 289iT^{2} \)
19 \( 1 + 21.7T + 361T^{2} \)
23 \( 1 + (1.77 + 1.77i)T + 529iT^{2} \)
29 \( 1 - 28.0T + 841T^{2} \)
31 \( 1 + 17.2iT - 961T^{2} \)
37 \( 1 + (6.50 + 6.50i)T + 1.36e3iT^{2} \)
41 \( 1 + 26.7T + 1.68e3T^{2} \)
43 \( 1 + (-33.1 + 33.1i)T - 1.84e3iT^{2} \)
47 \( 1 + (-18.5 - 18.5i)T + 2.20e3iT^{2} \)
53 \( 1 + (-48.3 - 48.3i)T + 2.80e3iT^{2} \)
59 \( 1 - 29.6iT - 3.48e3T^{2} \)
61 \( 1 + 21.0iT - 3.72e3T^{2} \)
67 \( 1 + (32.4 + 32.4i)T + 4.48e3iT^{2} \)
71 \( 1 - 16.0iT - 5.04e3T^{2} \)
73 \( 1 + (-57.3 - 57.3i)T + 5.32e3iT^{2} \)
79 \( 1 - 75.8iT - 6.24e3T^{2} \)
83 \( 1 + (-51.9 + 51.9i)T - 6.88e3iT^{2} \)
89 \( 1 - 174. iT - 7.92e3T^{2} \)
97 \( 1 + (16.6 - 16.6i)T - 9.40e3iT^{2} \)
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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

−13.79555767013381708788236909876, −12.46878125553842883238638634864, −11.95856616758321110568912022591, −11.02685788620364851239012983390, −8.958614996450881933714277800428, −8.237041892639130579162612370061, −7.20566728872118239609741023652, −6.40315074342101693263290385781, −4.22477815554135633892250999252, −2.34730008857012980853675029065, 0.69441233491994569917528715843, 3.53438050103299432174730216890, 4.83160063783118934216517813793, 6.06354016854790123068234386086, 8.313813022354988603622733823404, 8.718159598571372894813343841126, 10.41802395397084741362698157420, 10.83101485191203102685721216455, 11.65052288502401838284224639242, 13.34319271497122349213222750142

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