Properties

Label 2-1900-19.7-c1-0-11
Degree $2$
Conductor $1900$
Sign $0.988 + 0.154i$
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  + (0.354 + 0.614i)3-s − 3.11·7-s + (1.24 − 2.16i)9-s − 3.52·11-s + (0.200 − 0.347i)13-s + (1.74 + 3.02i)17-s + (4.35 − 0.251i)19-s + (−1.10 − 1.91i)21-s + (−3.65 + 6.33i)23-s + 3.89·27-s + (3.96 − 6.86i)29-s + 5.73·31-s + (−1.25 − 2.16i)33-s + 10.5·37-s + 0.284·39-s + ⋯
L(s)  = 1  + (0.204 + 0.354i)3-s − 1.17·7-s + (0.416 − 0.720i)9-s − 1.06·11-s + (0.0556 − 0.0964i)13-s + (0.424 + 0.734i)17-s + (0.998 − 0.0577i)19-s + (−0.240 − 0.416i)21-s + (−0.762 + 1.32i)23-s + 0.750·27-s + (0.736 − 1.27i)29-s + 1.03·31-s + (−0.217 − 0.377i)33-s + 1.73·37-s + 0.0456·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.154i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $0.988 + 0.154i$
Analytic conductor: \(15.1715\)
Root analytic conductor: \(3.89507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (501, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1/2),\ 0.988 + 0.154i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.540434752\)
\(L(\frac12)\) \(\approx\) \(1.540434752\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.354 - 0.614i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + 3.11T + 7T^{2} \)
11 \( 1 + 3.52T + 11T^{2} \)
13 \( 1 + (-0.200 + 0.347i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.74 - 3.02i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (3.65 - 6.33i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.96 + 6.86i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.73T + 31T^{2} \)
37 \( 1 - 10.5T + 37T^{2} \)
41 \( 1 + (-0.555 - 0.961i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.30 + 7.45i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.76 + 6.51i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.27 + 9.14i)T + (-26.5 - 45.8i)T^{2} \)
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 \( 1 + (-4.20 + 7.28i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.31 - 7.46i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.870 - 1.50i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.50 - 7.80i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 4.07T + 83T^{2} \)
89 \( 1 + (-6.61 + 11.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.85 + 6.67i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.475988045201510672537150034273, −8.372027837600520029165797601229, −7.71776233741541259011463828582, −6.76084602689752310765820347864, −6.02107801128035269379891311758, −5.23390445351944933615647532594, −3.98967341623659089336374605859, −3.40625936076382101601544270712, −2.44151761711381527914359986969, −0.73745666631595858320859644218, 0.923946432030445081735854405248, 2.56196201597221664960880897501, 2.99013568645125816463351888773, 4.37034934102219645039670220685, 5.17781349316233292236589043941, 6.16393142436777416405855118341, 6.91630836169900169623530093402, 7.70810909875635184219479250394, 8.269113829575844184175722470675, 9.337037437209996908046949611387

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