Properties

Label 2-3450-5.4-c1-0-11
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 − 1.27·11-s + i·12-s − 3.27i·13-s − 3·14-s + 16-s − 2.27i·17-s i·18-s − 3.27·19-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 − 0.384·11-s + 0.288i·12-s − 0.908i·13-s − 0.801·14-s + 0.250·16-s − 0.551i·17-s − 0.235i·18-s − 0.751·19-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

\(L(1)\) \(\approx\) \(1.180871086\)
\(L(\frac12)\) \(\approx\) \(1.180871086\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.27T + 11T^{2} \)
13 \( 1 + 3.27iT - 13T^{2} \)
17 \( 1 + 2.27iT - 17T^{2} \)
19 \( 1 + 3.27T + 19T^{2} \)
29 \( 1 - 9.54T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 - 10.8iT - 37T^{2} \)
41 \( 1 + 2.72T + 41T^{2} \)
43 \( 1 + 7.27iT - 43T^{2} \)
47 \( 1 - 8.27iT - 47T^{2} \)
53 \( 1 - 10.5iT - 53T^{2} \)
59 \( 1 + 12.5T + 59T^{2} \)
61 \( 1 - 0.549T + 61T^{2} \)
67 \( 1 + 4.54iT - 67T^{2} \)
71 \( 1 + 8.82T + 71T^{2} \)
73 \( 1 - 15.5iT - 73T^{2} \)
79 \( 1 - 11.8T + 79T^{2} \)
83 \( 1 - 2.45iT - 83T^{2} \)
89 \( 1 + 8.27T + 89T^{2} \)
97 \( 1 - 14iT - 97T^{2} \)
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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

−8.465849024686079272715110420828, −8.221409177258851023081753942011, −7.37735380184961890147066826696, −6.46337790191808800033049415311, −6.02153536498621179639485852938, −5.17144636821831289427289938401, −4.55032043518056130799547198520, −3.06627010749682571024401609711, −2.53032087339082973469241448771, −1.07410897599337993229649042030, 0.40468141871978687200202296072, 1.71062271864153786691536149519, 2.77454304463137164584763367649, 3.75214447211088061700268482957, 4.37744605085685063959787984480, 4.91046855665867977372712755976, 6.13333043701568385410333876521, 6.79886654172621083134234808096, 7.79557791135833487919264537474, 8.496682801831264748782044265252

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