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Properties

Label 2-105-21.20-c3-0-31
Degree $2$
Conductor $105$
Sign $0.616 - 0.787i$
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  − 4.61i·2-s + (−3.98 − 3.33i)3-s − 13.2·4-s − 5·5-s + (−15.3 + 18.3i)6-s + (0.582 + 18.5i)7-s + 24.3i·8-s + (4.79 + 26.5i)9-s + 23.0i·10-s − 65.3i·11-s + (52.9 + 44.2i)12-s + 30.9i·13-s + (85.4 − 2.68i)14-s + (19.9 + 16.6i)15-s + 6.26·16-s − 85.8·17-s + ⋯
L(s)  = 1  − 1.63i·2-s + (−0.767 − 0.641i)3-s − 1.66·4-s − 0.447·5-s + (−1.04 + 1.25i)6-s + (0.0314 + 0.999i)7-s + 1.07i·8-s + (0.177 + 0.984i)9-s + 0.729i·10-s − 1.79i·11-s + (1.27 + 1.06i)12-s + 0.660i·13-s + (1.63 − 0.0513i)14-s + (0.343 + 0.286i)15-s + 0.0978·16-s − 1.22·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.616 - 0.787i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.616 - 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.616 - 0.787i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ 0.616 - 0.787i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0786760 + 0.0382992i\)
\(L(\frac12)\) \(\approx\) \(0.0786760 + 0.0382992i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.98 + 3.33i)T \)
5 \( 1 + 5T \)
7 \( 1 + (-0.582 - 18.5i)T \)
good2 \( 1 + 4.61iT - 8T^{2} \)
11 \( 1 + 65.3iT - 1.33e3T^{2} \)
13 \( 1 - 30.9iT - 2.19e3T^{2} \)
17 \( 1 + 85.8T + 4.91e3T^{2} \)
19 \( 1 + 6.24iT - 6.85e3T^{2} \)
23 \( 1 - 193. iT - 1.21e4T^{2} \)
29 \( 1 - 30.7iT - 2.43e4T^{2} \)
31 \( 1 + 61.0iT - 2.97e4T^{2} \)
37 \( 1 + 8.71T + 5.06e4T^{2} \)
41 \( 1 + 387.T + 6.89e4T^{2} \)
43 \( 1 + 281.T + 7.95e4T^{2} \)
47 \( 1 + 208.T + 1.03e5T^{2} \)
53 \( 1 + 211. iT - 1.48e5T^{2} \)
59 \( 1 + 213.T + 2.05e5T^{2} \)
61 \( 1 + 673. iT - 2.26e5T^{2} \)
67 \( 1 + 500.T + 3.00e5T^{2} \)
71 \( 1 + 319. iT - 3.57e5T^{2} \)
73 \( 1 + 555. iT - 3.89e5T^{2} \)
79 \( 1 - 842.T + 4.93e5T^{2} \)
83 \( 1 - 887.T + 5.71e5T^{2} \)
89 \( 1 + 480.T + 7.04e5T^{2} \)
97 \( 1 + 477. iT - 9.12e5T^{2} \)
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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

−11.92417198881976005882289394774, −11.50413990718433685651694436208, −10.85415676764904130514891679097, −9.294809705625033537730978689636, −8.296501963354670252692139224179, −6.44856488002341733884559976079, −5.06351775123610793475229695894, −3.34670194319871241613295899271, −1.78715047372656069361369965753, −0.05316000204138030792666177034, 4.29448337803343816912534659910, 4.89663307006279600320388717710, 6.55527728052814139113362275397, 7.16420084232880479964881169874, 8.462056876984018956211250983694, 9.839443608633965039503486447464, 10.76683700178118052330973375250, 12.23787078229924554516646562317, 13.35118868907728582384063343864, 14.80680317288135634656262555501

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