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Properties

Label 2-1900-19.11-c1-0-29
Degree $2$
Conductor $1900$
Sign $-0.965 + 0.260i$
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  + (1.28 − 2.22i)3-s − 3.56·7-s + (−1.79 − 3.11i)9-s + 4.56·11-s + (−0.5 − 0.866i)13-s + (2.86 − 4.96i)17-s + (−4.35 + 0.221i)19-s + (−4.58 + 7.93i)21-s + (−2.79 − 4.84i)23-s − 1.53·27-s + (−3.38 − 5.86i)29-s + 4.59·31-s + (5.86 − 10.1i)33-s − 3.56·37-s − 2.56·39-s + ⋯
L(s)  = 1  + (0.741 − 1.28i)3-s − 1.34·7-s + (−0.599 − 1.03i)9-s + 1.37·11-s + (−0.138 − 0.240i)13-s + (0.695 − 1.20i)17-s + (−0.998 + 0.0507i)19-s + (−1.00 + 1.73i)21-s + (−0.583 − 1.01i)23-s − 0.296·27-s + (−0.628 − 1.08i)29-s + 0.826·31-s + (1.02 − 1.76i)33-s − 0.586·37-s − 0.411·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.260i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.260i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $-0.965 + 0.260i$
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: \((2,\ 1900,\ (\ :1/2),\ -0.965 + 0.260i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.453084989\)
\(L(\frac12)\) \(\approx\) \(1.453084989\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.35 - 0.221i)T \)
good3 \( 1 + (-1.28 + 2.22i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + 3.56T + 7T^{2} \)
11 \( 1 - 4.56T + 11T^{2} \)
13 \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.86 + 4.96i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (2.79 + 4.84i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.38 + 5.86i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.59T + 31T^{2} \)
37 \( 1 + 3.56T + 37T^{2} \)
41 \( 1 + (5.35 - 9.27i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.65 - 9.79i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.38 + 5.86i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.70 + 2.94i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.38 + 5.86i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.06 + 7.04i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.16 + 5.48i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.515 + 0.892i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.36 - 11.0i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.43 - 4.21i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 9.40T + 83T^{2} \)
89 \( 1 + (6.13 + 10.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.31 + 2.27i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.668717953031433293825460597749, −8.120718929498164426694350621107, −7.18187902380849937413893935064, −6.45599904977895177033512156126, −6.24313513703636065764435645008, −4.66356741350235350019302111880, −3.49247154071924598252203591105, −2.81643761718053788957588773708, −1.75176449239221694799046164055, −0.46265124338586867138336770447, 1.76754437669870928657413430627, 3.22912309140527595862660727979, 3.68524862860445388582995943454, 4.30263282830594545345469012481, 5.55945949772664959900287889151, 6.38432692367769168659495932369, 7.13795640679142310097897488112, 8.389557942037752863694751864290, 8.998807582453479284075549577938, 9.457837194328594200072107489487

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