Properties

Label 2-3450-5.4-c1-0-55
Degree $2$
Conductor $3450$
Sign $-0.894 + 0.447i$
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 + i·8-s − 9-s + 2·11-s i·12-s + 16-s − 6i·17-s + i·18-s − 4·19-s − 2i·22-s + i·23-s − 24-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s + 0.353i·8-s − 0.333·9-s + 0.603·11-s − 0.288i·12-s + 0.250·16-s − 1.45i·17-s + 0.235i·18-s − 0.917·19-s − 0.426i·22-s + 0.208i·23-s − 0.204·24-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3450\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 23\)
Sign: $-0.894 + 0.447i$
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: \((2,\ 3450,\ (\ :1/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7264555001\)
\(L(\frac12)\) \(\approx\) \(0.7264555001\)
\(L(\frac{3}{2})\) 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 \( 1 - 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 2iT - 43T^{2} \)
47 \( 1 + 4iT - 47T^{2} \)
53 \( 1 + 2iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 2iT - 67T^{2} \)
71 \( 1 + 10T + 71T^{2} \)
73 \( 1 + 10iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 + 4T + 89T^{2} \)
97 \( 1 + 16iT - 97T^{2} \)
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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

−8.671708303761309882348714750303, −7.57687396507234789301980145130, −6.82413785473615188906291682597, −5.83156591356276053829149837625, −5.03493033654997584626407117887, −4.31560799559786495352568316536, −3.53292860288006180194309143443, −2.69948330882332794981554872726, −1.64447752652188570346616505164, −0.22235180309532984340696959816, 1.30215581999366280544888684773, 2.30796909921570149753098045218, 3.67155124745838427436242287520, 4.24475847355889004858784986094, 5.38805820420280999529783973466, 6.07897520738141816143513653911, 6.63068410472462867595353760249, 7.40359581359802599289776427613, 8.078056624496485883520350704103, 8.800841106865568606248544608588

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