Properties

Label 2-1900-95.94-c0-0-0
Degree $2$
Conductor $1900$
Sign $-0.447 - 0.894i$
Analytic cond. $0.948223$
Root an. cond. $0.973767$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·7-s − 9-s − 11-s + i·17-s − 19-s + 2i·23-s i·43-s + i·47-s − 61-s i·63-s i·73-s i·77-s + 81-s + 2i·83-s + 99-s + ⋯
L(s)  = 1  + i·7-s − 9-s − 11-s + i·17-s − 19-s + 2i·23-s i·43-s + i·47-s − 61-s i·63-s i·73-s i·77-s + 81-s + 2i·83-s + 99-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(0.948223\)
Root analytic conductor: \(0.973767\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :0),\ -0.447 - 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6738358016\)
\(L(\frac12)\) \(\approx\) \(0.6738358016\)
\(L(1)\) 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 \( 1 + T^{2} \)
7 \( 1 - iT - T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 - iT - T^{2} \)
23 \( 1 - 2iT - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + iT - T^{2} \)
47 \( 1 - iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - 2iT - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.509947449414602419106130512195, −8.809579887284687448615259578896, −8.193609026008999431328878211945, −7.48196833966580025376413301999, −6.22018702024616175722806379509, −5.70871104286145118615489737517, −5.01508359892013033273867357364, −3.73575783513243868403494954662, −2.78733877447800069076163046380, −1.90230738564628567184693902590, 0.45808780948722940640938766709, 2.31783756224388070910610549951, 3.10376437912904939628873921552, 4.32457160612193697710962987326, 4.98472888269810639769922012330, 6.02048931963967397356330763431, 6.79472724693543134209112575982, 7.63875873177797577440868630389, 8.364594750470749039077062346468, 9.013980968579495021598026174935

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