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Properties

Label 2-105-35.18-c2-0-0
Degree $2$
Conductor $105$
Sign $0.0726 - 0.997i$
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  + (−3.56 − 0.955i)2-s + (−1.67 + 0.448i)3-s + (8.33 + 4.81i)4-s + (−1.05 − 4.88i)5-s + 6.39·6-s + (−6.28 − 3.07i)7-s + (−14.6 − 14.6i)8-s + (2.59 − 1.50i)9-s + (−0.890 + 18.4i)10-s + (−4.09 + 7.09i)11-s + (−16.0 − 4.31i)12-s + (14.0 + 14.0i)13-s + (19.4 + 16.9i)14-s + (3.96 + 7.70i)15-s + (19.0 + 32.9i)16-s + (1.81 + 6.76i)17-s + ⋯
L(s)  = 1  + (−1.78 − 0.477i)2-s + (−0.557 + 0.149i)3-s + (2.08 + 1.20i)4-s + (−0.211 − 0.977i)5-s + 1.06·6-s + (−0.898 − 0.439i)7-s + (−1.83 − 1.83i)8-s + (0.288 − 0.166i)9-s + (−0.0890 + 1.84i)10-s + (−0.372 + 0.644i)11-s + (−1.34 − 0.359i)12-s + (1.08 + 1.08i)13-s + (1.39 + 1.21i)14-s + (0.264 + 0.513i)15-s + (1.18 + 2.06i)16-s + (0.106 + 0.397i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0726 - 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0726 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.0726 - 0.997i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (88, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ 0.0726 - 0.997i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.144324 + 0.134190i\)
\(L(\frac12)\) \(\approx\) \(0.144324 + 0.134190i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.67 - 0.448i)T \)
5 \( 1 + (1.05 + 4.88i)T \)
7 \( 1 + (6.28 + 3.07i)T \)
good2 \( 1 + (3.56 + 0.955i)T + (3.46 + 2i)T^{2} \)
11 \( 1 + (4.09 - 7.09i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-14.0 - 14.0i)T + 169iT^{2} \)
17 \( 1 + (-1.81 - 6.76i)T + (-250. + 144.5i)T^{2} \)
19 \( 1 + (18.2 - 10.5i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (8.43 - 31.4i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 - 22.1iT - 841T^{2} \)
31 \( 1 + (-13.7 + 23.7i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-14.9 - 4.01i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 - 0.496T + 1.68e3T^{2} \)
43 \( 1 + (33.4 + 33.4i)T + 1.84e3iT^{2} \)
47 \( 1 + (-24.1 - 6.46i)T + (1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (34.3 - 9.20i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (15.5 + 9.00i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-13.4 - 23.3i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (1.44 + 5.37i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 105.T + 5.04e3T^{2} \)
73 \( 1 + (81.6 - 21.8i)T + (4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (86.5 - 49.9i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-95.6 - 95.6i)T + 6.88e3iT^{2} \)
89 \( 1 + (-54.3 + 31.3i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-59.0 + 59.0i)T - 9.40e3iT^{2} \)
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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

−13.28544632418972481687585166997, −12.31715370882094674873212086201, −11.42523024650146404582987867347, −10.35108790646142807420057023831, −9.529458093011853806384557847356, −8.625155292851558274619942008952, −7.42884370281941824229517705453, −6.19354946535361654807561337130, −3.93666689331913444543272244773, −1.50962308950228602743294467948, 0.27995665422362412495114019522, 2.79260735813472857357774783054, 6.04943437927188563517964260364, 6.53114742927717602894968991547, 7.85358453253258254190026015584, 8.802730331955009912946715007711, 10.23718083703956371468836032372, 10.66637088196189260670219556168, 11.69813023983345558229164409923, 13.18159020530643279897509084454

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