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Properties

Label 2-105-105.89-c1-0-11
Degree $2$
Conductor $105$
Sign $-0.707 + 0.706i$
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  + (1.25 − 2.17i)2-s + (−1.51 − 0.831i)3-s + (−2.14 − 3.71i)4-s + (−0.00874 + 2.23i)5-s + (−3.71 + 2.25i)6-s + (1.65 − 2.06i)7-s − 5.74·8-s + (1.61 + 2.52i)9-s + (4.84 + 2.82i)10-s + (1.48 − 0.859i)11-s + (0.172 + 7.42i)12-s − 0.360·13-s + (−2.39 − 6.18i)14-s + (1.87 − 3.39i)15-s + (−2.91 + 5.04i)16-s + (−2.20 + 1.27i)17-s + ⋯
L(s)  = 1  + (0.886 − 1.53i)2-s + (−0.877 − 0.479i)3-s + (−1.07 − 1.85i)4-s + (−0.00391 + 0.999i)5-s + (−1.51 + 0.922i)6-s + (0.627 − 0.778i)7-s − 2.03·8-s + (0.539 + 0.841i)9-s + (1.53 + 0.892i)10-s + (0.448 − 0.259i)11-s + (0.0496 + 2.14i)12-s − 0.100·13-s + (−0.639 − 1.65i)14-s + (0.483 − 0.875i)15-s + (−0.727 + 1.26i)16-s + (−0.534 + 0.308i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.706i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.707 + 0.706i$
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: \((2,\ 105,\ (\ :1/2),\ -0.707 + 0.706i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.463484 - 1.11943i\)
\(L(\frac12)\) \(\approx\) \(0.463484 - 1.11943i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.51 + 0.831i)T \)
5 \( 1 + (0.00874 - 2.23i)T \)
7 \( 1 + (-1.65 + 2.06i)T \)
good2 \( 1 + (-1.25 + 2.17i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (-1.48 + 0.859i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.360T + 13T^{2} \)
17 \( 1 + (2.20 - 1.27i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.93 - 2.84i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.25 - 2.17i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.76iT - 29T^{2} \)
31 \( 1 + (2.41 - 1.39i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.86 - 1.65i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.63T + 41T^{2} \)
43 \( 1 - 10.0iT - 43T^{2} \)
47 \( 1 + (5.04 + 2.91i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.727 - 1.25i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.42 + 5.93i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.38 - 0.801i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.15 - 1.24i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 13.1iT - 71T^{2} \)
73 \( 1 + (5.82 + 10.0i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.93 + 12.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.50iT - 83T^{2} \)
89 \( 1 + (6.10 - 10.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 8.18T + 97T^{2} \)
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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

−13.29676455233457042911024526351, −12.06416332220459471662282874559, −11.34043945160925083066020921924, −10.73977837477490526133975889476, −9.838340546362846965126939103375, −7.57783481482205819518985813094, −6.20624422915053150953830413195, −4.85825841414254918795286846099, −3.49760979400510120025705046538, −1.61506508586070255501949679055, 4.23388862116935021532450829441, 5.08863068416277351139204112172, 5.86733288485789977751464652833, 7.18324297851113474922798415165, 8.531809890153955474472289367078, 9.484942624337857521719070083662, 11.52172476789027222010754573945, 12.25410129748318876467641583612, 13.23945873407069073773054887222, 14.39836013028000797235831583634

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