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Properties

Label 2-1900-19.7-c1-0-12
Degree $2$
Conductor $1900$
Sign $-0.0133 + 0.999i$
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  + (−1.43 − 2.48i)3-s + 3.54·7-s + (−2.62 + 4.54i)9-s − 1.81·11-s + (−1.60 + 2.78i)13-s + (3.99 + 6.92i)17-s + (−0.863 − 4.27i)19-s + (−5.09 − 8.82i)21-s + (4.21 − 7.30i)23-s + 6.46·27-s + (4.29 − 7.43i)29-s − 1.70·31-s + (2.60 + 4.51i)33-s + 5.50·37-s + 9.23·39-s + ⋯
L(s)  = 1  + (−0.829 − 1.43i)3-s + 1.34·7-s + (−0.875 + 1.51i)9-s − 0.547·11-s + (−0.445 + 0.771i)13-s + (0.969 + 1.67i)17-s + (−0.198 − 0.980i)19-s + (−1.11 − 1.92i)21-s + (0.878 − 1.52i)23-s + 1.24·27-s + (0.796 − 1.38i)29-s − 0.306·31-s + (0.453 + 0.786i)33-s + 0.905·37-s + 1.47·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0133 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0133 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $-0.0133 + 0.999i$
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: \((2,\ 1900,\ (\ :1/2),\ -0.0133 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.425424620\)
\(L(\frac12)\) \(\approx\) \(1.425424620\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.863 + 4.27i)T \)
good3 \( 1 + (1.43 + 2.48i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 - 3.54T + 7T^{2} \)
11 \( 1 + 1.81T + 11T^{2} \)
13 \( 1 + (1.60 - 2.78i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.99 - 6.92i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-4.21 + 7.30i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.29 + 7.43i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.70T + 31T^{2} \)
37 \( 1 - 5.50T + 37T^{2} \)
41 \( 1 + (-4.05 - 7.02i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.51 - 4.35i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.674 + 1.16i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.10 + 1.92i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.960 + 1.66i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.83 + 4.90i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.64 + 8.04i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.94 + 5.09i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.63 + 2.82i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.08 + 3.61i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.30T + 83T^{2} \)
89 \( 1 + (2.73 - 4.73i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.99 + 6.91i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.649784044094214128421104348614, −7.998000709588401809479938334960, −7.55303794232749770505658545498, −6.53765028564678155058298769402, −6.05371143188561447485862330080, −4.99818957669171132103285332161, −4.41438033596843843145591064442, −2.58942475032762611756702580178, −1.77576403229660320455935181679, −0.75085873550615587350148426439, 1.02637183534941131803959385383, 2.77112325581464928045796028204, 3.75750970139935277572892923622, 4.77159739921791275835224454197, 5.37112826745683840521696722395, 5.58826027077960622302576610082, 7.23703219919263531450419633825, 7.77801396632961530638967654368, 8.824898154730425322978286024423, 9.572099550669515683270477055287

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