Properties

Label 2-3450-5.4-c1-0-62
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 − 4i·7-s i·8-s − 9-s − 2·11-s + i·12-s + 4·14-s + 16-s − 2i·17-s i·18-s − 4·21-s − 2i·22-s + i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s − 1.51i·7-s − 0.353i·8-s − 0.333·9-s − 0.603·11-s + 0.288i·12-s + 1.06·14-s + 0.250·16-s − 0.485i·17-s − 0.235i·18-s − 0.872·21-s − 0.426i·22-s + 0.208i·23-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

\(L(1)\) \(\approx\) \(0.6826453941\)
\(L(\frac12)\) \(\approx\) \(0.6826453941\)
\(L(\frac{3}{2})\) 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 \( 1 + 4iT - 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 10iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 2iT - 43T^{2} \)
47 \( 1 + 12iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 + 14T + 61T^{2} \)
67 \( 1 + 2iT - 67T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 8iT - 83T^{2} \)
89 \( 1 - 8T + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−7.889023824608979117311818396722, −7.54491682378360765865229958244, −6.94609393656072801726459869821, −6.19594758454851536533325490384, −5.33060200352736723320980304836, −4.49330831004622263226039870989, −3.72083375007140684079021589082, −2.64156845921894809913343354082, −1.24595322760943312660261276553, −0.21512056250361237315298819939, 1.57849636336488535884124184564, 2.69721655858785123884089056511, 3.09173744627403608460008293702, 4.36831178494353495923880787683, 4.95330297041993171173896061820, 5.80110155840610961592399105441, 6.36392212570722653978851548852, 7.76860115865797413948480623479, 8.369749158830849622726163695797, 9.048310987488586818230960361768

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