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Properties

Label 2-3900-65.49-c1-0-4
Degree $2$
Conductor $3900$
Sign $-0.452 - 0.891i$
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 + (−1.25 + 2.18i)7-s + (0.499 − 0.866i)9-s + (0.590 − 0.341i)11-s + (−1.86 − 3.08i)13-s + (4.15 + 2.39i)17-s + (−5.65 − 3.26i)19-s + 2.51i·21-s + (−6.38 + 3.68i)23-s − 0.999i·27-s + (2.31 + 4.00i)29-s − 0.600i·31-s + (0.341 − 0.590i)33-s + (5.11 + 8.85i)37-s + (−3.15 − 1.74i)39-s + ⋯
L(s)  = 1  + (0.499 − 0.288i)3-s + (−0.475 + 0.824i)7-s + (0.166 − 0.288i)9-s + (0.178 − 0.102i)11-s + (−0.516 − 0.856i)13-s + (1.00 + 0.581i)17-s + (−1.29 − 0.748i)19-s + 0.549i·21-s + (−1.33 + 0.768i)23-s − 0.192i·27-s + (0.429 + 0.744i)29-s − 0.107i·31-s + (0.0593 − 0.102i)33-s + (0.840 + 1.45i)37-s + (−0.505 − 0.279i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.452 - 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.452 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3900\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.452 - 0.891i$
Analytic conductor: \(31.1416\)
Root analytic conductor: \(5.58047\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3900} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3900,\ (\ :1/2),\ -0.452 - 0.891i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9806335776\)
\(L(\frac12)\) \(\approx\) \(0.9806335776\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.86 + 3.08i)T \)
good7 \( 1 + (1.25 - 2.18i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.590 + 0.341i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-4.15 - 2.39i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.65 + 3.26i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.38 - 3.68i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.31 - 4.00i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.600iT - 31T^{2} \)
37 \( 1 + (-5.11 - 8.85i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.31 - 0.758i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.01 + 3.47i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 4.73T + 47T^{2} \)
53 \( 1 - 10.3iT - 53T^{2} \)
59 \( 1 + (8.19 + 4.73i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.66 - 9.80i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.20 - 3.81i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.291 - 0.168i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 4.50T + 73T^{2} \)
79 \( 1 - 9.19T + 79T^{2} \)
83 \( 1 - 10.0T + 83T^{2} \)
89 \( 1 + (11.2 - 6.51i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.60 + 7.97i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.631492894864398700910173150086, −8.047334040538178502347066356398, −7.38656598234719744761503212037, −6.36914570662124287823313649635, −5.92623463826615260863474468400, −5.01449430016316377298169062774, −4.00239911310648149686752087358, −3.08460193905535883102096024950, −2.46768925028397159785840439102, −1.33770681510930163874736823224, 0.25400047846197663319517796683, 1.78192940933152717353123161699, 2.64752730754933436821075862061, 3.83495670815393786230864191404, 4.13387004752419413981992526457, 5.08683836795389355660976402601, 6.22688751491823616852912442476, 6.70055383248932638522176060312, 7.70914820195487567488678822563, 8.062871911020344052841458837625

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