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Properties

Label 2-105-105.89-c1-0-1
Degree $2$
Conductor $105$
Sign $-0.288 - 0.957i$
Analytic cond. $0.838429$
Root an. cond. $0.915657$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.757 + 1.31i)2-s + (−1.24 − 1.20i)3-s + (−0.147 − 0.254i)4-s + (1.42 + 1.72i)5-s + (2.52 − 0.722i)6-s + (−0.0753 + 2.64i)7-s − 2.58·8-s + (0.102 + 2.99i)9-s + (−3.33 + 0.570i)10-s + (−1.86 + 1.07i)11-s + (−0.123 + 0.494i)12-s + 3.48·13-s + (−3.41 − 2.10i)14-s + (0.291 − 3.86i)15-s + (2.25 − 3.89i)16-s + (3.09 − 1.78i)17-s + ⋯
L(s)  = 1  + (−0.535 + 0.927i)2-s + (−0.719 − 0.694i)3-s + (−0.0735 − 0.127i)4-s + (0.638 + 0.769i)5-s + (1.02 − 0.294i)6-s + (−0.0284 + 0.999i)7-s − 0.913·8-s + (0.0341 + 0.999i)9-s + (−1.05 + 0.180i)10-s + (−0.560 + 0.323i)11-s + (−0.0356 + 0.142i)12-s + 0.965·13-s + (−0.911 − 0.561i)14-s + (0.0753 − 0.997i)15-s + (0.562 − 0.974i)16-s + (0.751 − 0.433i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.288 - 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.288 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.288 - 0.957i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ -0.288 - 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.408453 + 0.549825i\)
\(L(\frac12)\) \(\approx\) \(0.408453 + 0.549825i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.24 + 1.20i)T \)
5 \( 1 + (-1.42 - 1.72i)T \)
7 \( 1 + (0.0753 - 2.64i)T \)
good2 \( 1 + (0.757 - 1.31i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (1.86 - 1.07i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.48T + 13T^{2} \)
17 \( 1 + (-3.09 + 1.78i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.05 + 0.611i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.757 + 1.31i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.95iT - 29T^{2} \)
31 \( 1 + (-2.75 + 1.58i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.75 + 3.90i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 11.8T + 41T^{2} \)
43 \( 1 - 2.99iT - 43T^{2} \)
47 \( 1 + (5.28 + 3.05i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.61 - 9.72i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.08 - 1.87i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.94 + 1.69i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.93 + 5.15i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.3iT - 71T^{2} \)
73 \( 1 + (-3.42 - 5.93i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.941 + 1.63i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9.10iT - 83T^{2} \)
89 \( 1 + (0.889 - 1.54i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.32T + 97T^{2} \)
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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

−14.19826071888011299395440074107, −13.00531017923393065320610239465, −12.00777510151906130218713178748, −10.97001400832301017213675433634, −9.644735426041410834955237603653, −8.334103983376273797642532655661, −7.29358999219339440217306589071, −6.22619674300028953409358436741, −5.57630470224097938725195773744, −2.57724965062916931086420493435, 1.11907301871462742327858604131, 3.60429434857794805111339540123, 5.24990784786118804858062457416, 6.36696154125094776981614464732, 8.453768895758370927039269282091, 9.562007192163686986634457213679, 10.42567017179429833304132883486, 10.96065108377255111896712225696, 12.19529381358550635186924914013, 13.15112210480602010228785370041

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