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Properties

Label 2-3450-5.4-c1-0-3
Degree $2$
Conductor $3450$
Sign $-0.447 + 0.894i$
Analytic cond. $27.5483$
Root an. cond. $5.24865$
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  + i·2-s + i·3-s − 4-s − 6-s + 3i·7-s − i·8-s − 9-s + 5.77·11-s − i·12-s − 3.77i·13-s − 3·14-s + 16-s − 0.772i·17-s − i·18-s − 7.77·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s + 1.13i·7-s − 0.353i·8-s − 0.333·9-s + 1.74·11-s − 0.288i·12-s − 1.04i·13-s − 0.801·14-s + 0.250·16-s − 0.187i·17-s − 0.235i·18-s − 1.78·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3450\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 23\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(27.5483\)
Root analytic conductor: \(5.24865\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3450} (2899, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3450,\ (\ :1/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5118395098\)
\(L(\frac12)\) \(\approx\) \(0.5118395098\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 - iT \)
5 \( 1 \)
23 \( 1 - iT \)
good7 \( 1 - 3iT - 7T^{2} \)
11 \( 1 - 5.77T + 11T^{2} \)
13 \( 1 + 3.77iT - 13T^{2} \)
17 \( 1 + 0.772iT - 17T^{2} \)
19 \( 1 + 7.77T + 19T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 + 9.54T + 31T^{2} \)
37 \( 1 + 6.77iT - 37T^{2} \)
41 \( 1 + 5.77T + 41T^{2} \)
43 \( 1 - 7.77iT - 43T^{2} \)
47 \( 1 - 8.77iT - 47T^{2} \)
53 \( 1 - 4iT - 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 + 9.54T + 61T^{2} \)
67 \( 1 - 9.54iT - 67T^{2} \)
71 \( 1 + 4.77T + 71T^{2} \)
73 \( 1 - 6.54iT - 73T^{2} \)
79 \( 1 + 2.22T + 79T^{2} \)
83 \( 1 + iT - 83T^{2} \)
89 \( 1 + 16.7T + 89T^{2} \)
97 \( 1 + 17.5iT - 97T^{2} \)
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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

−9.106195671368235450562193067695, −8.511419931753141191184040482817, −7.65644381648191668841671567717, −6.73117219500119443167790146138, −5.94633900042620774772133921625, −5.58703683027526901376214404631, −4.50112493664960132739156434598, −3.86543782610498259709703700050, −2.87995449836068819897366693085, −1.63564762629728883461267356740, 0.14879498556167005692433561203, 1.51967853135372507620019725834, 1.95647343829635660082877364090, 3.53036774613882834292210671603, 3.99311842864866784368020487581, 4.71209872952103865229188774091, 6.01726224873405442689463826884, 6.77271638952846823927352434794, 7.12305219079943851541664147012, 8.260782531072619542126453899734

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