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Properties

Label 2-3900-13.10-c1-0-12
Degree $2$
Conductor $3900$
Sign $0.565 - 0.824i$
Analytic cond. $31.1416$
Root an. cond. $5.58047$
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  + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)9-s + (−0.633 + 0.366i)11-s + (−0.232 + 3.59i)13-s + (3.09 − 5.36i)17-s + (−1.96 − 1.13i)19-s + (2.36 + 4.09i)23-s + 0.999·27-s + (−0.267 − 0.464i)29-s + 2.26i·31-s + (0.633 + 0.366i)33-s + (−9.92 + 5.73i)37-s + (3.23 − 1.59i)39-s + (−3.63 + 2.09i)41-s + (5.19 − 9i)43-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.166 + 0.288i)9-s + (−0.191 + 0.110i)11-s + (−0.0643 + 0.997i)13-s + (0.751 − 1.30i)17-s + (−0.450 − 0.260i)19-s + (0.493 + 0.854i)23-s + 0.192·27-s + (−0.0497 − 0.0861i)29-s + 0.407i·31-s + (0.110 + 0.0637i)33-s + (−1.63 + 0.942i)37-s + (0.517 − 0.255i)39-s + (−0.567 + 0.327i)41-s + (0.792 − 1.37i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.565 - 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.565 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3900\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.565 - 0.824i$
Analytic conductor: \(31.1416\)
Root analytic conductor: \(5.58047\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3900} (2701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3900,\ (\ :1/2),\ 0.565 - 0.824i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.221920224\)
\(L(\frac12)\) \(\approx\) \(1.221920224\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 \)
13 \( 1 + (0.232 - 3.59i)T \)
good7 \( 1 + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.633 - 0.366i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.09 + 5.36i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.96 + 1.13i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.36 - 4.09i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.267 + 0.464i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.26iT - 31T^{2} \)
37 \( 1 + (9.92 - 5.73i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.63 - 2.09i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.19 + 9i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 10.7iT - 47T^{2} \)
53 \( 1 - 11.1T + 53T^{2} \)
59 \( 1 + (6.46 + 3.73i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.767 - 1.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.66 + 3.26i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.29 - 3.63i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 14.6iT - 73T^{2} \)
79 \( 1 - 3.53T + 79T^{2} \)
83 \( 1 - 13.8iT - 83T^{2} \)
89 \( 1 + (14.6 - 8.46i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.03 + 0.598i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.566915912429311597178052431315, −7.74414467732089395673389687626, −6.95189213839669190229636027510, −6.67951676325563618191873112215, −5.43248512655145610512660420682, −5.07668267734311666524581005555, −4.00129974170401153366244512189, −3.01859353476352280969019355301, −2.06741053988039637159924062180, −1.02447075461639494447631887865, 0.42216963008231358817432633642, 1.79312462964953995064392478718, 3.00112302406036522404887851653, 3.72574084880730358930376434165, 4.57210549424601273942465371720, 5.48926150437049839847622746220, 5.93254144710971937105288836820, 6.83414542746911399741314127813, 7.74319423430390144546918212292, 8.391322565640317762726624640187

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