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Properties

Label 2-1900-95.64-c1-0-0
Degree $2$
Conductor $1900$
Sign $-0.776 - 0.629i$
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  + (1.73 − i)3-s + 4i·7-s + (0.499 − 0.866i)9-s − 3·11-s + (−5.19 − 3i)13-s + (−1.73 + i)17-s + (−3.5 + 2.59i)19-s + (4 + 6.92i)21-s + (−3.46 − 2i)23-s + 4.00i·27-s + (0.5 − 0.866i)29-s − 5·31-s + (−5.19 + 3i)33-s + 4i·37-s − 12·39-s + ⋯
L(s)  = 1  + (0.999 − 0.577i)3-s + 1.51i·7-s + (0.166 − 0.288i)9-s − 0.904·11-s + (−1.44 − 0.832i)13-s + (−0.420 + 0.242i)17-s + (−0.802 + 0.596i)19-s + (0.872 + 1.51i)21-s + (−0.722 − 0.417i)23-s + 0.769i·27-s + (0.0928 − 0.160i)29-s − 0.898·31-s + (−0.904 + 0.522i)33-s + 0.657i·37-s − 1.92·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.776 - 0.629i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.776 - 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $-0.776 - 0.629i$
Analytic conductor: \(15.1715\)
Root analytic conductor: \(3.89507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1/2),\ -0.776 - 0.629i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6731027595\)
\(L(\frac12)\) \(\approx\) \(0.6731027595\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.5 - 2.59i)T \)
good3 \( 1 + (-1.73 + i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 - 4iT - 7T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + (5.19 + 3i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.73 - i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (3.46 + 2i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 5T + 31T^{2} \)
37 \( 1 - 4iT - 37T^{2} \)
41 \( 1 + (1 + 1.73i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.19 - 3i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.19 - 3i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (12.1 + 7i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.5 + 12.9i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-10.3 + 6i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 16iT - 83T^{2} \)
89 \( 1 + (-8.5 + 14.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (10.3 - 6i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.268141541131674455709499398873, −8.647772666898519425740432926298, −7.957138119741025855223310450802, −7.53034830920199410380035957486, −6.33037633272993282165111450303, −5.51834184769869826589510585644, −4.77308700900269260062626641040, −3.32294012882191153578081542509, −2.38117468400252566111854659720, −2.12914862500964996315940573592, 0.18587966166460704873263648031, 2.07558944369751907460501090615, 2.92295046690702591822291711862, 4.11532198377796116124187267334, 4.37702454940207645751945942305, 5.52780305382413688666017398860, 6.96420295599280626332818855167, 7.27180968168562485018063620025, 8.175462151746250403522212491123, 8.969623525322633000142345214524

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