Properties

Label 2-105-105.104-c3-0-37
Degree $2$
Conductor $105$
Sign $0.839 + 0.543i$
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  + 5.11·2-s + (−1.51 − 4.96i)3-s + 18.1·4-s + (10.9 + 2.25i)5-s + (−7.75 − 25.4i)6-s + (−11.4 + 14.5i)7-s + 51.7·8-s + (−22.3 + 15.0i)9-s + (55.9 + 11.5i)10-s − 55.4i·11-s + (−27.5 − 90.0i)12-s − 20.9·13-s + (−58.6 + 74.3i)14-s + (−5.42 − 57.8i)15-s + 119.·16-s + 96.8i·17-s + ⋯
L(s)  = 1  + 1.80·2-s + (−0.292 − 0.956i)3-s + 2.26·4-s + (0.979 + 0.201i)5-s + (−0.527 − 1.72i)6-s + (−0.619 + 0.785i)7-s + 2.28·8-s + (−0.829 + 0.558i)9-s + (1.77 + 0.364i)10-s − 1.51i·11-s + (−0.661 − 2.16i)12-s − 0.447·13-s + (−1.11 + 1.41i)14-s + (−0.0933 − 0.995i)15-s + 1.86·16-s + 1.38i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.839 + 0.543i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.839 + 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.839 + 0.543i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ 0.839 + 0.543i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.92096 - 1.15799i\)
\(L(\frac12)\) \(\approx\) \(3.92096 - 1.15799i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.51 + 4.96i)T \)
5 \( 1 + (-10.9 - 2.25i)T \)
7 \( 1 + (11.4 - 14.5i)T \)
good2 \( 1 - 5.11T + 8T^{2} \)
11 \( 1 + 55.4iT - 1.33e3T^{2} \)
13 \( 1 + 20.9T + 2.19e3T^{2} \)
17 \( 1 - 96.8iT - 4.91e3T^{2} \)
19 \( 1 + 33.1iT - 6.85e3T^{2} \)
23 \( 1 + 111.T + 1.21e4T^{2} \)
29 \( 1 - 156. iT - 2.43e4T^{2} \)
31 \( 1 - 80.4iT - 2.97e4T^{2} \)
37 \( 1 + 180. iT - 5.06e4T^{2} \)
41 \( 1 + 36.5T + 6.89e4T^{2} \)
43 \( 1 - 52.5iT - 7.95e4T^{2} \)
47 \( 1 + 259. iT - 1.03e5T^{2} \)
53 \( 1 - 191.T + 1.48e5T^{2} \)
59 \( 1 - 705.T + 2.05e5T^{2} \)
61 \( 1 + 427. iT - 2.26e5T^{2} \)
67 \( 1 + 306. iT - 3.00e5T^{2} \)
71 \( 1 - 513. iT - 3.57e5T^{2} \)
73 \( 1 + 360.T + 3.89e5T^{2} \)
79 \( 1 - 85.9T + 4.93e5T^{2} \)
83 \( 1 + 886. iT - 5.71e5T^{2} \)
89 \( 1 + 1.41e3T + 7.04e5T^{2} \)
97 \( 1 - 1.05e3T + 9.12e5T^{2} \)
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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

−13.12290927160941818760955967137, −12.58817649403329515538784082238, −11.56009223321651725216245731585, −10.52727676253206112118378398337, −8.596496217884043999306718872678, −6.82132053070338809061312802372, −5.98778002068121960964736712750, −5.45673676846181837675234291362, −3.26534195475109435053125590816, −2.08527548704675770204588262163, 2.54843974325741529001088802838, 4.12399331651657799565421463544, 4.95903143769330045744242273114, 6.08104374150967472487688056558, 7.17124992618384320746669459779, 9.700877086993782569284439776117, 10.17234380210433699046536792454, 11.62337963435198229724700148148, 12.49574730185969622649848906084, 13.53259271673583953458174848119

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