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Properties

Label 2-1900-95.18-c1-0-4
Degree $2$
Conductor $1900$
Sign $-0.992 + 0.123i$
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  + (−2.25 + 2.25i)3-s + (1.48 − 1.48i)7-s − 7.15i·9-s + 0.806·11-s + (0.437 − 0.437i)13-s + (−3.15 + 3.15i)17-s + (1.81 − 3.96i)19-s + 6.67i·21-s + (−1.86 − 1.86i)23-s + (9.36 + 9.36i)27-s − 4.50·29-s + 6.67i·31-s + (−1.81 + 1.81i)33-s + (−5.29 − 5.29i)37-s + 1.96i·39-s + ⋯
L(s)  = 1  + (−1.30 + 1.30i)3-s + (0.559 − 0.559i)7-s − 2.38i·9-s + 0.243·11-s + (0.121 − 0.121i)13-s + (−0.765 + 0.765i)17-s + (0.416 − 0.909i)19-s + 1.45i·21-s + (−0.389 − 0.389i)23-s + (1.80 + 1.80i)27-s − 0.836·29-s + 1.19i·31-s + (−0.316 + 0.316i)33-s + (−0.870 − 0.870i)37-s + 0.315i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.123i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 + 0.123i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $-0.992 + 0.123i$
Analytic conductor: \(15.1715\)
Root analytic conductor: \(3.89507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1/2),\ -0.992 + 0.123i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3683636613\)
\(L(\frac12)\) \(\approx\) \(0.3683636613\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.81 + 3.96i)T \)
good3 \( 1 + (2.25 - 2.25i)T - 3iT^{2} \)
7 \( 1 + (-1.48 + 1.48i)T - 7iT^{2} \)
11 \( 1 - 0.806T + 11T^{2} \)
13 \( 1 + (-0.437 + 0.437i)T - 13iT^{2} \)
17 \( 1 + (3.15 - 3.15i)T - 17iT^{2} \)
23 \( 1 + (1.86 + 1.86i)T + 23iT^{2} \)
29 \( 1 + 4.50T + 29T^{2} \)
31 \( 1 - 6.67iT - 31T^{2} \)
37 \( 1 + (5.29 + 5.29i)T + 37iT^{2} \)
41 \( 1 - 11.1iT - 41T^{2} \)
43 \( 1 + (-3.86 - 3.86i)T + 43iT^{2} \)
47 \( 1 + (6.83 - 6.83i)T - 47iT^{2} \)
53 \( 1 + (8.92 - 8.92i)T - 53iT^{2} \)
59 \( 1 - 12.9T + 59T^{2} \)
61 \( 1 + 2.15T + 61T^{2} \)
67 \( 1 + (4.94 + 4.94i)T + 67iT^{2} \)
71 \( 1 + 3.04iT - 71T^{2} \)
73 \( 1 + (-6.19 - 6.19i)T + 73iT^{2} \)
79 \( 1 + 1.46T + 79T^{2} \)
83 \( 1 + (5.32 + 5.32i)T + 83iT^{2} \)
89 \( 1 + 14.2T + 89T^{2} \)
97 \( 1 + (7.46 + 7.46i)T + 97iT^{2} \)
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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

−9.697232117124205763202640341170, −9.121219673992720554227961950658, −8.159668050538412261173760417486, −7.01834753022634188078917144145, −6.29196671503035327372377924123, −5.50468689578102485707635458015, −4.60917237602102744019479613519, −4.23950934693747837060004723411, −3.15809670252812594555794392376, −1.31032539207231209814891767580, 0.17042453017812554007639641078, 1.57766763939987853950963319573, 2.26215148521439123216773014333, 3.92083145855649559848557827191, 5.25132424505412176214656035530, 5.48808694059663425890324357160, 6.49190599005265750843432875623, 7.08388287006537095075633783750, 7.86359266611962422937066586375, 8.568384679112737243932468162948

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