Properties

Label 2-105-21.5-c3-0-7
Degree $2$
Conductor $105$
Sign $0.925 - 0.378i$
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  + (−3.31 − 1.91i)2-s + (4.42 + 2.72i)3-s + (3.33 + 5.77i)4-s + (−2.5 + 4.33i)5-s + (−9.46 − 17.5i)6-s + (8.94 − 16.2i)7-s + 5.08i·8-s + (12.1 + 24.0i)9-s + (16.5 − 9.57i)10-s + (−12.9 + 7.47i)11-s + (−0.963 + 34.6i)12-s + 63.8i·13-s + (−60.7 + 36.6i)14-s + (−22.8 + 12.3i)15-s + (36.4 − 63.0i)16-s + (−11.0 − 19.0i)17-s + ⋯
L(s)  = 1  + (−1.17 − 0.677i)2-s + (0.851 + 0.523i)3-s + (0.417 + 0.722i)4-s + (−0.223 + 0.387i)5-s + (−0.644 − 1.19i)6-s + (0.483 − 0.875i)7-s + 0.224i·8-s + (0.451 + 0.892i)9-s + (0.524 − 0.302i)10-s + (−0.355 + 0.204i)11-s + (−0.0231 + 0.833i)12-s + 1.36i·13-s + (−1.15 + 0.699i)14-s + (−0.393 + 0.212i)15-s + (0.569 − 0.985i)16-s + (−0.156 − 0.271i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 - 0.378i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.925 - 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.925 - 0.378i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ 0.925 - 0.378i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.08503 + 0.213439i\)
\(L(\frac12)\) \(\approx\) \(1.08503 + 0.213439i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.42 - 2.72i)T \)
5 \( 1 + (2.5 - 4.33i)T \)
7 \( 1 + (-8.94 + 16.2i)T \)
good2 \( 1 + (3.31 + 1.91i)T + (4 + 6.92i)T^{2} \)
11 \( 1 + (12.9 - 7.47i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 63.8iT - 2.19e3T^{2} \)
17 \( 1 + (11.0 + 19.0i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-137. - 79.2i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-162. - 93.8i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 179. iT - 2.43e4T^{2} \)
31 \( 1 + (-64.0 + 37.0i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (182. - 316. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 168.T + 6.89e4T^{2} \)
43 \( 1 + 60.4T + 7.95e4T^{2} \)
47 \( 1 + (5.05 - 8.74i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (467. - 269. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (165. + 286. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (15.5 + 8.98i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (111. + 193. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 563. iT - 3.57e5T^{2} \)
73 \( 1 + (-525. + 303. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-88.0 + 152. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 635.T + 5.71e5T^{2} \)
89 \( 1 + (-388. + 673. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.03e3iT - 9.12e5T^{2} \)
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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

−13.67239171371368294170831537700, −11.74150111047905137746667208477, −10.98595130787455919448491898376, −9.939757368360947081061757039433, −9.318238317382877498623037826808, −8.025455713494681182154693149223, −7.29795793354418239459364369193, −4.77151772225633441159336579964, −3.21615647598186928917587371718, −1.57543115064913054102323605609, 0.937086527274140469361951117283, 3.01661093140845132631402031526, 5.33842457962595715492019426160, 6.99256535866651389951907571003, 7.928920307744551298132027627358, 8.691642685888949555115527406284, 9.385945660654780731047661827210, 10.83201088364230843379753104547, 12.40865309700869582254334595651, 13.11221234182060245176757966648

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