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Properties

Label 2-1900-19.11-c1-0-26
Degree $2$
Conductor $1900$
Sign $-0.813 + 0.582i$
Analytic cond. $15.1715$
Root an. cond. $3.89507$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)3-s + (1 + 1.73i)9-s − 4·11-s + (−0.5 − 0.866i)13-s + (1.5 − 2.59i)17-s + (−4 + 1.73i)19-s + (2.5 + 4.33i)23-s − 5·27-s + (−3.5 − 6.06i)29-s + 4·31-s + (2 − 3.46i)33-s − 10·37-s + 0.999·39-s + (2.5 − 4.33i)41-s + (−2.5 + 4.33i)43-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (0.333 + 0.577i)9-s − 1.20·11-s + (−0.138 − 0.240i)13-s + (0.363 − 0.630i)17-s + (−0.917 + 0.397i)19-s + (0.521 + 0.902i)23-s − 0.962·27-s + (−0.649 − 1.12i)29-s + 0.718·31-s + (0.348 − 0.603i)33-s − 1.64·37-s + 0.160·39-s + (0.390 − 0.676i)41-s + (−0.381 + 0.660i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 + 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.813 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $-0.813 + 0.582i$
Analytic conductor: \(15.1715\)
Root analytic conductor: \(3.89507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1900,\ (\ :1/2),\ -0.813 + 0.582i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (4 - 1.73i)T \)
good3 \( 1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-2.5 - 4.33i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.5 + 6.06i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + 10T + 37T^{2} \)
41 \( 1 + (-2.5 + 4.33i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.5 - 4.33i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.5 + 6.06i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.5 - 9.52i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.5 + 9.52i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.5 + 2.59i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.5 - 9.52i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-7.5 + 12.9i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.5 + 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (1.5 + 2.59i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.5 - 4.33i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.989845845342367437544805239775, −7.892524891917832966917134819022, −7.59791217356223092350716492494, −6.45199098482668909269739373495, −5.40615849177093872873670141514, −5.00032343592065568304053243122, −3.99667377812366721334553729347, −2.91987037007810720725395987440, −1.84338344060154728953582166081, 0, 1.45625333627236327551263550352, 2.60799685690575793086005394328, 3.68531751020570680582832074051, 4.75450861290282967168848324743, 5.53763941380112885393628704764, 6.55663909393840842003513273988, 6.98322702247328319463089078027, 7.989570482257657290675379671721, 8.622435830875364603478764477064

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