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Properties

Label 2-200-8.5-c1-0-8
Degree $2$
Conductor $200$
Sign $0.707 - 0.707i$
Analytic cond. $1.59700$
Root an. cond. $1.26372$
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  + (1.36 − 0.366i)2-s + 2.73i·3-s + (1.73 − i)4-s + (1 + 3.73i)6-s − 0.732·7-s + (1.99 − 2i)8-s − 4.46·9-s + 2i·11-s + (2.73 + 4.73i)12-s − 3.46i·13-s + (−1 + 0.267i)14-s + (1.99 − 3.46i)16-s − 3.46·17-s + (−6.09 + 1.63i)18-s − 0.535i·19-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s + 1.57i·3-s + (0.866 − 0.5i)4-s + (0.408 + 1.52i)6-s − 0.276·7-s + (0.707 − 0.707i)8-s − 1.48·9-s + 0.603i·11-s + (0.788 + 1.36i)12-s − 0.960i·13-s + (−0.267 + 0.0716i)14-s + (0.499 − 0.866i)16-s − 0.840·17-s + (−1.43 + 0.385i)18-s − 0.122i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(1.59700\)
Root analytic conductor: \(1.26372\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{200} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 200,\ (\ :1/2),\ 0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82310 + 0.755153i\)
\(L(\frac12)\) \(\approx\) \(1.82310 + 0.755153i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 + 0.366i)T \)
5 \( 1 \)
good3 \( 1 - 2.73iT - 3T^{2} \)
7 \( 1 + 0.732T + 7T^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 + 3.46T + 17T^{2} \)
19 \( 1 + 0.535iT - 19T^{2} \)
23 \( 1 - 6.19T + 23T^{2} \)
29 \( 1 + 6.92iT - 29T^{2} \)
31 \( 1 + 5.46T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 - 1.46T + 41T^{2} \)
43 \( 1 + 5.26iT - 43T^{2} \)
47 \( 1 + 3.26T + 47T^{2} \)
53 \( 1 - 11.4iT - 53T^{2} \)
59 \( 1 - 7.46iT - 59T^{2} \)
61 \( 1 - 8.92iT - 61T^{2} \)
67 \( 1 - 10.7iT - 67T^{2} \)
71 \( 1 - 5.46T + 71T^{2} \)
73 \( 1 + 7.46T + 73T^{2} \)
79 \( 1 + 1.07T + 79T^{2} \)
83 \( 1 - 1.26iT - 83T^{2} \)
89 \( 1 - 8.92T + 89T^{2} \)
97 \( 1 - 14.3T + 97T^{2} \)
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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

−12.67065498568686674699407571742, −11.43608905537050851892638114914, −10.67194239307554254274788599484, −9.945866771313705529504609703271, −8.972932572654809180346407228696, −7.28601238535297850720143320124, −5.85041551936445313113891560737, −4.87015517522962115371896364688, −3.97055997904480693761724235852, −2.78268952173905654153624522446, 1.83342274361543785051074145781, 3.27991297599157474048785793937, 5.02882779130563354887572436271, 6.42725474082146945134677209836, 6.83291220485538776692749363331, 7.926355947701997810716764479918, 9.013758836363201580201155147901, 11.01471456700969895498128039764, 11.61325157540304703107878508825, 12.77696721521154162488637899374

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