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Properties

Label 2-200-40.37-c2-0-23
Degree $2$
Conductor $200$
Sign $0.138 + 0.990i$
Analytic cond. $5.44960$
Root an. cond. $2.33443$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.689 + 1.87i)2-s + (−1.90 − 1.90i)3-s + (−3.04 + 2.58i)4-s + (2.25 − 4.88i)6-s + (−0.633 − 0.633i)7-s + (−6.96 − 3.93i)8-s − 1.77i·9-s + 0.688i·11-s + (10.7 + 0.874i)12-s + (−14.5 − 14.5i)13-s + (0.752 − 1.62i)14-s + (2.59 − 15.7i)16-s + (−18.4 − 18.4i)17-s + (3.32 − 1.22i)18-s + 29.9·19-s + ⋯
L(s)  = 1  + (0.344 + 0.938i)2-s + (−0.633 − 0.633i)3-s + (−0.762 + 0.647i)4-s + (0.376 − 0.813i)6-s + (−0.0905 − 0.0905i)7-s + (−0.870 − 0.492i)8-s − 0.196i·9-s + 0.0626i·11-s + (0.893 + 0.0728i)12-s + (−1.12 − 1.12i)13-s + (0.0537 − 0.116i)14-s + (0.162 − 0.986i)16-s + (−1.08 − 1.08i)17-s + (0.184 − 0.0678i)18-s + 1.57·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.138 + 0.990i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.138 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $0.138 + 0.990i$
Analytic conductor: \(5.44960\)
Root analytic conductor: \(2.33443\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{200} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 200,\ (\ :1),\ 0.138 + 0.990i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.516590 - 0.449156i\)
\(L(\frac12)\) \(\approx\) \(0.516590 - 0.449156i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.689 - 1.87i)T \)
5 \( 1 \)
good3 \( 1 + (1.90 + 1.90i)T + 9iT^{2} \)
7 \( 1 + (0.633 + 0.633i)T + 49iT^{2} \)
11 \( 1 - 0.688iT - 121T^{2} \)
13 \( 1 + (14.5 + 14.5i)T + 169iT^{2} \)
17 \( 1 + (18.4 + 18.4i)T + 289iT^{2} \)
19 \( 1 - 29.9T + 361T^{2} \)
23 \( 1 + (-17.4 + 17.4i)T - 529iT^{2} \)
29 \( 1 + 14.9T + 841T^{2} \)
31 \( 1 + 41.8T + 961T^{2} \)
37 \( 1 + (37.1 - 37.1i)T - 1.36e3iT^{2} \)
41 \( 1 - 0.334T + 1.68e3T^{2} \)
43 \( 1 + (2.74 + 2.74i)T + 1.84e3iT^{2} \)
47 \( 1 + (-32.7 - 32.7i)T + 2.20e3iT^{2} \)
53 \( 1 + (-11.6 - 11.6i)T + 2.80e3iT^{2} \)
59 \( 1 + 17.4T + 3.48e3T^{2} \)
61 \( 1 - 38.9iT - 3.72e3T^{2} \)
67 \( 1 + (-52.9 + 52.9i)T - 4.48e3iT^{2} \)
71 \( 1 + 72.3T + 5.04e3T^{2} \)
73 \( 1 + (10.9 - 10.9i)T - 5.32e3iT^{2} \)
79 \( 1 - 76.8iT - 6.24e3T^{2} \)
83 \( 1 + (24.2 + 24.2i)T + 6.88e3iT^{2} \)
89 \( 1 + 150. iT - 7.92e3T^{2} \)
97 \( 1 + (15.3 + 15.3i)T + 9.40e3iT^{2} \)
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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

−12.28243007316252030639363216809, −11.36027533526483711357519077339, −9.837980810447958519896959399760, −8.869376368619172708325234582062, −7.37020693840696974688224765336, −7.04228616175986334482394101052, −5.69261425126353145116594582075, −4.88478521026484657066846055262, −3.12031679532543290994576628498, −0.36925708666712252441213906540, 2.01977552701459476907358483373, 3.74903287675166841032586151936, 4.85711139984287748838986247703, 5.68757317893553772024365290700, 7.26313087215517858219758309899, 8.999168167300674132928803405223, 9.680167335947873432988281156774, 10.74823045666291384297109466988, 11.37178585320949481132243439130, 12.22644468723898556232411038869

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