Properties

Label 2-1900-5.4-c1-0-0
Degree $2$
Conductor $1900$
Sign $0.447 - 0.894i$
Analytic cond. $15.1715$
Root an. cond. $3.89507$
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  − 2.73i·3-s + 2i·7-s − 4.46·9-s − 3.46·11-s + 2.73i·13-s − 3.46i·17-s − 19-s + 5.46·21-s + 3.46i·23-s + 3.99i·27-s − 3.46·29-s − 1.46·31-s + 9.46i·33-s + 6.73i·37-s + 7.46·39-s + ⋯
L(s)  = 1  − 1.57i·3-s + 0.755i·7-s − 1.48·9-s − 1.04·11-s + 0.757i·13-s − 0.840i·17-s − 0.229·19-s + 1.19·21-s + 0.722i·23-s + 0.769i·27-s − 0.643·29-s − 0.262·31-s + 1.64i·33-s + 1.10i·37-s + 1.19·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(15.1715\)
Root analytic conductor: \(3.89507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (1749, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6613437874\)
\(L(\frac12)\) \(\approx\) \(0.6613437874\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
19 \( 1 + T \)
good3 \( 1 + 2.73iT - 3T^{2} \)
7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 + 3.46T + 11T^{2} \)
13 \( 1 - 2.73iT - 13T^{2} \)
17 \( 1 + 3.46iT - 17T^{2} \)
23 \( 1 - 3.46iT - 23T^{2} \)
29 \( 1 + 3.46T + 29T^{2} \)
31 \( 1 + 1.46T + 31T^{2} \)
37 \( 1 - 6.73iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 4.92iT - 43T^{2} \)
47 \( 1 - 12.9iT - 47T^{2} \)
53 \( 1 - 10.7iT - 53T^{2} \)
59 \( 1 + 6.92T + 59T^{2} \)
61 \( 1 - 12.3T + 61T^{2} \)
67 \( 1 - 6.73iT - 67T^{2} \)
71 \( 1 + 2.53T + 71T^{2} \)
73 \( 1 - 0.535iT - 73T^{2} \)
79 \( 1 + 2.92T + 79T^{2} \)
83 \( 1 + 3.46iT - 83T^{2} \)
89 \( 1 - 15.4T + 89T^{2} \)
97 \( 1 + 16.5iT - 97T^{2} \)
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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

−9.151553390492859181763123869064, −8.421033077023396698866351859550, −7.63185697457803680906269437462, −7.14411658315026498063411798209, −6.22584884157680453077185216869, −5.58943526163631367155100796194, −4.62423123865531631987689892022, −3.04697598017794262047488864882, −2.32410381523485315122514870649, −1.35632934320536687730304894410, 0.23667438150447221158205953482, 2.28078474052169429511839132782, 3.54608357388191831612022758817, 3.94757589520230145568180429970, 5.07764536967559769974554854520, 5.45895404721001998588141621011, 6.65122693251984489817794257360, 7.69998132180816987453692019551, 8.414523565307635688484526316401, 9.162139958254118214005248897832

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