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Properties

Label 2-1900-95.49-c1-0-0
Degree $2$
Conductor $1900$
Sign $-0.579 - 0.814i$
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  + (−0.614 − 0.354i)3-s − 3.11i·7-s + (−1.24 − 2.16i)9-s − 3.52·11-s + (0.347 − 0.200i)13-s + (3.02 + 1.74i)17-s + (−4.35 − 0.251i)19-s + (−1.10 + 1.91i)21-s + (−6.33 + 3.65i)23-s + 3.89i·27-s + (−3.96 − 6.86i)29-s + 5.73·31-s + (2.16 + 1.25i)33-s + 10.5i·37-s − 0.284·39-s + ⋯
L(s)  = 1  + (−0.354 − 0.204i)3-s − 1.17i·7-s + (−0.416 − 0.720i)9-s − 1.06·11-s + (0.0964 − 0.0556i)13-s + (0.734 + 0.424i)17-s + (−0.998 − 0.0577i)19-s + (−0.240 + 0.416i)21-s + (−1.32 + 0.762i)23-s + 0.750i·27-s + (−0.736 − 1.27i)29-s + 1.03·31-s + (0.377 + 0.217i)33-s + 1.73i·37-s − 0.0456·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.579 - 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.579 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $-0.579 - 0.814i$
Analytic conductor: \(15.1715\)
Root analytic conductor: \(3.89507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1/2),\ -0.579 - 0.814i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.05955791068\)
\(L(\frac12)\) \(\approx\) \(0.05955791068\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.35 + 0.251i)T \)
good3 \( 1 + (0.614 + 0.354i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + 3.11iT - 7T^{2} \)
11 \( 1 + 3.52T + 11T^{2} \)
13 \( 1 + (-0.347 + 0.200i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.02 - 1.74i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (6.33 - 3.65i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.96 + 6.86i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.73T + 31T^{2} \)
37 \( 1 - 10.5iT - 37T^{2} \)
41 \( 1 + (-0.555 + 0.961i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-7.45 - 4.30i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.51 - 3.76i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-9.14 + 5.27i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.25 - 7.37i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.61 - 7.98i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.28 - 4.20i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.31 + 7.46i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.50 + 0.870i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.50 - 7.80i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 4.07iT - 83T^{2} \)
89 \( 1 + (6.61 + 11.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.67 + 3.85i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.802259551611845920615280751196, −8.482308756049808408378935284119, −7.910201917815807941118383435567, −7.17832030508042193395015284412, −6.18925456843793445270073727446, −5.71247636278169326125557381619, −4.47496846121515429255161846231, −3.76545744545909322579022640939, −2.67458216416585306700427232678, −1.22497531527715149419935323470, 0.02332856883384785183001659625, 2.11713202619889180535601515894, 2.70543550216386405026950234969, 4.04533845817237560510644080443, 5.15727702186776146663397263769, 5.55902011738360405772165527419, 6.33897577653234911404360073567, 7.53275192771163021661160506481, 8.232617128708554727486048873908, 8.824872786833215883575550621686

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