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Properties

Label 2-1900-19.11-c1-0-28
Degree $2$
Conductor $1900$
Sign $-0.889 + 0.455i$
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  + (1.21 − 2.10i)3-s − 0.663·7-s + (−1.45 − 2.52i)9-s − 1.80·11-s + (−1.15 − 1.99i)13-s + (2.18 − 3.77i)17-s + (4.21 − 1.12i)19-s + (−0.806 + 1.39i)21-s + (−1.04 − 1.81i)23-s + 0.216·27-s + (−0.974 − 1.68i)29-s − 9.52·31-s + (−2.19 + 3.80i)33-s − 2.97·37-s − 5.60·39-s + ⋯
L(s)  = 1  + (0.701 − 1.21i)3-s − 0.250·7-s + (−0.485 − 0.840i)9-s − 0.545·11-s + (−0.319 − 0.553i)13-s + (0.528 − 0.915i)17-s + (0.966 − 0.257i)19-s + (−0.176 + 0.304i)21-s + (−0.218 − 0.378i)23-s + 0.0416·27-s + (−0.180 − 0.313i)29-s − 1.71·31-s + (−0.382 + 0.663i)33-s − 0.489·37-s − 0.896·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.889 + 0.455i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.889 + 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $-0.889 + 0.455i$
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: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1/2),\ -0.889 + 0.455i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.603087092\)
\(L(\frac12)\) \(\approx\) \(1.603087092\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.21 + 1.12i)T \)
good3 \( 1 + (-1.21 + 2.10i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + 0.663T + 7T^{2} \)
11 \( 1 + 1.80T + 11T^{2} \)
13 \( 1 + (1.15 + 1.99i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.18 + 3.77i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.04 + 1.81i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.974 + 1.68i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 9.52T + 31T^{2} \)
37 \( 1 + 2.97T + 37T^{2} \)
41 \( 1 + (0.247 - 0.428i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.93 + 6.81i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.28 + 5.69i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.15 + 1.99i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.88 - 6.73i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.36 + 9.28i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.29 - 3.96i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.95 - 5.12i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.80 - 4.86i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.99 + 5.19i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.20T + 83T^{2} \)
89 \( 1 + (-6.65 - 11.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.08 - 8.80i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.766157845329043816016820836244, −7.87646018830242356198009788179, −7.42423246184715226815886451081, −6.81737851696009222712365693006, −5.69901419551140304493055439728, −4.98504532156422374790693590650, −3.49447000999584816383576231628, −2.77634078885250290701435213947, −1.83808933556959734894152391541, −0.50960587103059906189934030604, 1.73843291023829082473626473155, 3.08312219900979702075891518995, 3.60770358549127678263991602040, 4.55936693591986889882191258007, 5.34596596789041187844360741781, 6.25255170852238253307568702270, 7.46563919993693154541061139606, 8.012143814197006536499336121536, 9.112221608717443027085027804170, 9.390243688039621978557835397692

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