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Properties

Label 2-105-105.104-c3-0-41
Degree $2$
Conductor $105$
Sign $-0.340 + 0.940i$
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  + 2.24·2-s + (1.06 − 5.08i)3-s − 2.96·4-s + (8.51 − 7.24i)5-s + (2.39 − 11.4i)6-s + (−12.2 − 13.8i)7-s − 24.6·8-s + (−24.7 − 10.8i)9-s + (19.1 − 16.2i)10-s + 25.7i·11-s + (−3.16 + 15.0i)12-s + 68.2·13-s + (−27.5 − 31.1i)14-s + (−27.7 − 51.0i)15-s − 31.5·16-s − 30.6i·17-s + ⋯
L(s)  = 1  + 0.793·2-s + (0.205 − 0.978i)3-s − 0.370·4-s + (0.761 − 0.648i)5-s + (0.162 − 0.776i)6-s + (−0.662 − 0.748i)7-s − 1.08·8-s + (−0.915 − 0.401i)9-s + (0.604 − 0.514i)10-s + 0.705i·11-s + (−0.0760 + 0.362i)12-s + 1.45·13-s + (−0.526 − 0.594i)14-s + (−0.478 − 0.878i)15-s − 0.492·16-s − 0.437i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.340 + 0.940i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.340 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.340 + 0.940i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ -0.340 + 0.940i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.18807 - 1.69430i\)
\(L(\frac12)\) \(\approx\) \(1.18807 - 1.69430i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.06 + 5.08i)T \)
5 \( 1 + (-8.51 + 7.24i)T \)
7 \( 1 + (12.2 + 13.8i)T \)
good2 \( 1 - 2.24T + 8T^{2} \)
11 \( 1 - 25.7iT - 1.33e3T^{2} \)
13 \( 1 - 68.2T + 2.19e3T^{2} \)
17 \( 1 + 30.6iT - 4.91e3T^{2} \)
19 \( 1 + 109. iT - 6.85e3T^{2} \)
23 \( 1 - 152.T + 1.21e4T^{2} \)
29 \( 1 - 191. iT - 2.43e4T^{2} \)
31 \( 1 + 16.4iT - 2.97e4T^{2} \)
37 \( 1 - 81.9iT - 5.06e4T^{2} \)
41 \( 1 - 372.T + 6.89e4T^{2} \)
43 \( 1 - 192. iT - 7.95e4T^{2} \)
47 \( 1 - 0.366iT - 1.03e5T^{2} \)
53 \( 1 + 5.95T + 1.48e5T^{2} \)
59 \( 1 + 198.T + 2.05e5T^{2} \)
61 \( 1 + 83.5iT - 2.26e5T^{2} \)
67 \( 1 + 1.08e3iT - 3.00e5T^{2} \)
71 \( 1 - 773. iT - 3.57e5T^{2} \)
73 \( 1 + 448.T + 3.89e5T^{2} \)
79 \( 1 + 1.27e3T + 4.93e5T^{2} \)
83 \( 1 + 429. iT - 5.71e5T^{2} \)
89 \( 1 - 5.29T + 7.04e5T^{2} \)
97 \( 1 - 435.T + 9.12e5T^{2} \)
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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

−13.12620655610912274977813395818, −12.57050021059277183372045886653, −11.09485417082365103465945061466, −9.420597951465135778316018502052, −8.737967193988919080056467415970, −7.03609686986348884887029333595, −6.03892184483393943376916696462, −4.71100113947551938041914422995, −3.06753601049456167526614350394, −0.958192213143347401177068476967, 2.93967443009370825280191612119, 3.87801570684189050280815125954, 5.67984534957956328562517141614, 6.06378054859482552277373421730, 8.544831447196488092645812780832, 9.302729470856920711178668863333, 10.38044718824338837127115747898, 11.45422656715603581001058856005, 12.91406061519081004921001878673, 13.70113351651372464896490018290

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