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Properties

Label 2-105-105.104-c1-0-7
Degree $2$
Conductor $105$
Sign $0.949 - 0.313i$
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  + 1.73·2-s + (1 + 1.41i)3-s + 0.999·4-s + (−1.73 − 1.41i)5-s + (1.73 + 2.44i)6-s + (−1 − 2.44i)7-s − 1.73·8-s + (−1.00 + 2.82i)9-s + (−2.99 − 2.44i)10-s + 2.82i·11-s + (0.999 + 1.41i)12-s + 4·13-s + (−1.73 − 4.24i)14-s + (0.267 − 3.86i)15-s − 5·16-s − 2.82i·17-s + ⋯
L(s)  = 1  + 1.22·2-s + (0.577 + 0.816i)3-s + 0.499·4-s + (−0.774 − 0.632i)5-s + (0.707 + 0.999i)6-s + (−0.377 − 0.925i)7-s − 0.612·8-s + (−0.333 + 0.942i)9-s + (−0.948 − 0.774i)10-s + 0.852i·11-s + (0.288 + 0.408i)12-s + 1.10·13-s + (−0.462 − 1.13i)14-s + (0.0691 − 0.997i)15-s − 1.25·16-s − 0.685i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.313i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.949 - 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.949 - 0.313i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ 0.949 - 0.313i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69185 + 0.271605i\)
\(L(\frac12)\) \(\approx\) \(1.69185 + 0.271605i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1 - 1.41i)T \)
5 \( 1 + (1.73 + 1.41i)T \)
7 \( 1 + (1 + 2.44i)T \)
good2 \( 1 - 1.73T + 2T^{2} \)
11 \( 1 - 2.82iT - 11T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 + 2.82iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 - 3.46T + 23T^{2} \)
29 \( 1 + 5.65iT - 29T^{2} \)
31 \( 1 - 9.79iT - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 3.46T + 41T^{2} \)
43 \( 1 + 4.89iT - 43T^{2} \)
47 \( 1 + 2.82iT - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 6.92T + 59T^{2} \)
61 \( 1 + 9.79iT - 61T^{2} \)
67 \( 1 - 4.89iT - 67T^{2} \)
71 \( 1 - 2.82iT - 71T^{2} \)
73 \( 1 + 8T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 2.82iT - 83T^{2} \)
89 \( 1 - 10.3T + 89T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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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

−13.72557509293963817535519269605, −13.11647508764656751678784743138, −11.96268792922019792594974489279, −10.81670185047741558198465204750, −9.501650525825397968690254685471, −8.442539820864108846324136547569, −6.99425896532514950415498876883, −5.14419943791786367584315889687, −4.22850298330842497974047409690, −3.35833776691349154454458590662, 2.87845056192373894188141371571, 3.77425723124847594775547346659, 5.82147678121895778841274969685, 6.61545921427657028036280746754, 8.188854155736163205269556742399, 9.059678842927342555093971210870, 11.11844524753782811890678657124, 11.94039509481470754801583399537, 12.91383202839479147883208731616, 13.57243873297159201483064218059

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