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Properties

Label 2-3450-5.4-c1-0-53
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s − i·3-s − 4-s + 6-s + 5i·7-s − i·8-s − 9-s − 3·11-s + i·12-s − 5i·13-s − 5·14-s + 16-s + 6i·17-s − i·18-s + 19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s + 1.88i·7-s − 0.353i·8-s − 0.333·9-s − 0.904·11-s + 0.288i·12-s − 1.38i·13-s − 1.33·14-s + 0.250·16-s + 1.45i·17-s − 0.235i·18-s + 0.229·19-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: \(1\)
Selberg data: \((2,\ 3450,\ (\ :1/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 5iT - 7T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + 5iT - 13T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 - T + 19T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 - 4iT - 37T^{2} \)
41 \( 1 + 7T + 41T^{2} \)
43 \( 1 - 7iT - 43T^{2} \)
47 \( 1 + 6iT - 47T^{2} \)
53 \( 1 + 8iT - 53T^{2} \)
59 \( 1 - 10T + 59T^{2} \)
61 \( 1 + 12T + 61T^{2} \)
67 \( 1 + 12iT - 67T^{2} \)
71 \( 1 - 10T + 71T^{2} \)
73 \( 1 + 15iT - 73T^{2} \)
79 \( 1 - 5T + 79T^{2} \)
83 \( 1 - 9iT - 83T^{2} \)
89 \( 1 + 14T + 89T^{2} \)
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

−8.115834380697886358168438477092, −7.975808290178836312794344463214, −6.70470720188722552524433288600, −6.08943052305313130257898917307, −5.39856908784494254563823646710, −5.04618632507053894991562266006, −3.46793342277986659998948611242, −2.70573007926049590274740718046, −1.72221470768633439696504255098, 0, 1.18612689804795668439847022152, 2.45544088445289748552781847035, 3.44978020616229464691262998044, 4.15625109956223619116024706452, 4.73840934580932392160571007672, 5.51785736810887445577064850860, 6.92402952944110218003659776481, 7.23373589747729263178783965254, 8.148375007545071543746432584432

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