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Properties

Label 2-105-35.4-c3-0-5
Degree $2$
Conductor $105$
Sign $0.00827 - 0.999i$
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  + (2.01 − 1.16i)2-s + (−2.59 − 1.5i)3-s + (−1.28 + 2.22i)4-s + (2.89 + 10.7i)5-s − 6.99·6-s + (−18.1 − 3.80i)7-s + 24.6i·8-s + (4.5 + 7.79i)9-s + (18.4 + 18.4i)10-s + (−17.6 + 30.6i)11-s + (6.67 − 3.85i)12-s + 8.12i·13-s + (−41.0 + 13.4i)14-s + (8.67 − 32.3i)15-s + (18.4 + 31.9i)16-s + (10.0 + 5.81i)17-s + ⋯
L(s)  = 1  + (0.713 − 0.411i)2-s + (−0.499 − 0.288i)3-s + (−0.160 + 0.278i)4-s + (0.258 + 0.965i)5-s − 0.475·6-s + (−0.978 − 0.205i)7-s + 1.08i·8-s + (0.166 + 0.288i)9-s + (0.582 + 0.582i)10-s + (−0.484 + 0.839i)11-s + (0.160 − 0.0926i)12-s + 0.173i·13-s + (−0.782 + 0.256i)14-s + (0.149 − 0.557i)15-s + (0.287 + 0.498i)16-s + (0.143 + 0.0829i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00827 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.00827 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.00827 - 0.999i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ 0.00827 - 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.909581 + 0.902081i\)
\(L(\frac12)\) \(\approx\) \(0.909581 + 0.902081i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.59 + 1.5i)T \)
5 \( 1 + (-2.89 - 10.7i)T \)
7 \( 1 + (18.1 + 3.80i)T \)
good2 \( 1 + (-2.01 + 1.16i)T + (4 - 6.92i)T^{2} \)
11 \( 1 + (17.6 - 30.6i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 8.12iT - 2.19e3T^{2} \)
17 \( 1 + (-10.0 - 5.81i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-36.3 - 62.9i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-144. + 83.3i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 259.T + 2.43e4T^{2} \)
31 \( 1 + (-112. + 194. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (311. - 179. i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 294.T + 6.89e4T^{2} \)
43 \( 1 - 167. iT - 7.95e4T^{2} \)
47 \( 1 + (53.3 - 30.8i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-180. - 104. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-329. + 571. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-310. - 537. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-88.6 - 51.2i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 747.T + 3.57e5T^{2} \)
73 \( 1 + (-556. - 321. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-403. - 698. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 660. iT - 5.71e5T^{2} \)
89 \( 1 + (-46.1 - 79.9i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 898. iT - 9.12e5T^{2} \)
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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

−13.24910269097033117794171557300, −12.69083683787307347751351701490, −11.60375245619219935072960610411, −10.58896528900862082875917762506, −9.540582340074582429639523917347, −7.71911666905310583834045559779, −6.67451461492559450169900683277, −5.37909662179722965911704336306, −3.79052270919079332195700129709, −2.48192099383414215632838670649, 0.60378209578500896891080140390, 3.55481687693243594801504607112, 5.16177955362453987730177201588, 5.67001087678059212600328925896, 6.99849468626884179528565116466, 8.930556853911083939353165814224, 9.672299857487131259731761633151, 10.90587954772527162078606496104, 12.36465275651582511920781649600, 13.14268930529944599156202551588

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