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Properties

Label 2-3900-65.4-c1-0-13
Degree $2$
Conductor $3900$
Sign $0.206 - 0.978i$
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  + (0.866 + 0.5i)3-s + (−1.80 − 3.12i)7-s + (0.499 + 0.866i)9-s + (3 + 1.73i)11-s + (3.46 + i)13-s + (−1.73 + i)17-s + (−6.12 + 3.53i)19-s − 3.60i·21-s + (−4.54 − 2.62i)23-s + 0.999i·27-s + (−3.62 + 6.27i)29-s + 3.46i·31-s + (1.73 + 3i)33-s + (−3.60 + 6.24i)37-s + (2.49 + 2.59i)39-s + ⋯
L(s)  = 1  + (0.499 + 0.288i)3-s + (−0.681 − 1.18i)7-s + (0.166 + 0.288i)9-s + (0.904 + 0.522i)11-s + (0.960 + 0.277i)13-s + (−0.420 + 0.242i)17-s + (−1.40 + 0.810i)19-s − 0.786i·21-s + (−0.947 − 0.546i)23-s + 0.192i·27-s + (−0.672 + 1.16i)29-s + 0.622i·31-s + (0.301 + 0.522i)33-s + (−0.592 + 1.02i)37-s + (0.400 + 0.416i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3900\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.206 - 0.978i$
Analytic conductor: \(31.1416\)
Root analytic conductor: \(5.58047\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3900} (2149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3900,\ (\ :1/2),\ 0.206 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.755009538\)
\(L(\frac12)\) \(\approx\) \(1.755009538\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 \)
13 \( 1 + (-3.46 - i)T \)
good7 \( 1 + (1.80 + 3.12i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3 - 1.73i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.73 - i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.12 - 3.53i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.54 + 2.62i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.62 - 6.27i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.46iT - 31T^{2} \)
37 \( 1 + (3.60 - 6.24i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6.24 - 3.60i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-7.14 + 4.12i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 10.6T + 47T^{2} \)
53 \( 1 - 9.24iT - 53T^{2} \)
59 \( 1 + (-4.62 + 2.66i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.0707 + 0.122i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.24 - 3.60i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 2.01T + 73T^{2} \)
79 \( 1 - 4.24T + 79T^{2} \)
83 \( 1 - 12.2T + 83T^{2} \)
89 \( 1 + (4.37 + 2.52i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.141 - 0.244i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.784127789073009752255646802133, −7.942076028250026331698581154318, −7.08148898416149535972644264683, −6.53782401929838199744733080078, −5.85509707291051446206126575236, −4.41478453420012188839232489100, −4.05025943937042607971242321493, −3.45338933078400592602289286537, −2.16178745225988462548747916472, −1.18309911426476412600389608181, 0.49521576890451753151884757586, 2.01795964143003541884810367220, 2.59842202112505902218479879231, 3.71697070630304631467563313209, 4.20571149140717296353442678528, 5.73289748081614545492058543782, 6.00945669249363779895462086921, 6.73005343866257531326711934244, 7.67759370777395911702824534957, 8.473591110065610101079257116183

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