Properties

Label 2-3450-5.4-c1-0-16
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 − 4·11-s + i·12-s + 6i·13-s + 16-s − 6i·17-s + i·18-s − 4·19-s + 4i·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 − 1.20·11-s + 0.288i·12-s + 1.66i·13-s + 0.250·16-s − 1.45i·17-s + 0.235i·18-s − 0.917·19-s + 0.852i·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

\(L(1)\) \(\approx\) \(1.189281772\)
\(L(\frac12)\) \(\approx\) \(1.189281772\)
\(L(\frac{3}{2})\) 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 \( 1 + 4T + 11T^{2} \)
13 \( 1 - 6iT - 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 - 14iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 - 12T + 79T^{2} \)
83 \( 1 - 16iT - 83T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
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

−8.747254709588423577340688029138, −7.77559413377405685528723924786, −7.11860118917362336569624195483, −6.39231239333070054480452787168, −5.35354256348786075248920017825, −4.67681722182461558600216132939, −3.80194338517480922719044684435, −2.55333156865305143807481482262, −2.19928422557990439415273858048, −0.823183893825295926897108568746, 0.47854328850114826098390521773, 2.24640206005644098778164929489, 3.28480431148627671517120141173, 4.08525513177826294622412090617, 5.01298742914042256728943913909, 5.66865917223226640709857301433, 6.17295461297151663563365913393, 7.30727041305469931317155355638, 8.081879826724826686626155483006, 8.342132617466661328431549257670

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