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Properties

Label 2-200-8.5-c1-0-6
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  + (−0.366 − 1.36i)2-s + 0.732i·3-s + (−1.73 + i)4-s + (1 − 0.267i)6-s + 2.73·7-s + (2 + 1.99i)8-s + 2.46·9-s − 2i·11-s + (−0.732 − 1.26i)12-s − 3.46i·13-s + (−1 − 3.73i)14-s + (1.99 − 3.46i)16-s + 3.46·17-s + (−0.901 − 3.36i)18-s + 7.46i·19-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)2-s + 0.422i·3-s + (−0.866 + 0.5i)4-s + (0.408 − 0.109i)6-s + 1.03·7-s + (0.707 + 0.707i)8-s + 0.821·9-s − 0.603i·11-s + (−0.211 − 0.366i)12-s − 0.960i·13-s + (−0.267 − 0.997i)14-s + (0.499 − 0.866i)16-s + 0.840·17-s + (−0.212 − 0.793i)18-s + 1.71i·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

\(L(1)\) \(\approx\) \(1.03382 - 0.428222i\)
\(L(\frac12)\) \(\approx\) \(1.03382 - 0.428222i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.366 + 1.36i)T \)
5 \( 1 \)
good3 \( 1 - 0.732iT - 3T^{2} \)
7 \( 1 - 2.73T + 7T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 - 3.46T + 17T^{2} \)
19 \( 1 - 7.46iT - 19T^{2} \)
23 \( 1 + 4.19T + 23T^{2} \)
29 \( 1 + 6.92iT - 29T^{2} \)
31 \( 1 - 1.46T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 5.46T + 41T^{2} \)
43 \( 1 - 8.73iT - 43T^{2} \)
47 \( 1 + 6.73T + 47T^{2} \)
53 \( 1 + 4.53iT - 53T^{2} \)
59 \( 1 + 0.535iT - 59T^{2} \)
61 \( 1 - 4.92iT - 61T^{2} \)
67 \( 1 + 7.26iT - 67T^{2} \)
71 \( 1 + 1.46T + 71T^{2} \)
73 \( 1 + 0.535T + 73T^{2} \)
79 \( 1 + 14.9T + 79T^{2} \)
83 \( 1 + 4.73iT - 83T^{2} \)
89 \( 1 + 4.92T + 89T^{2} \)
97 \( 1 + 6.39T + 97T^{2} \)
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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

−12.14369459864233846878152471173, −11.33230415362469380920239414876, −10.23011361036643184970825285948, −9.858588065064706849663044077977, −8.237881899898414674359498437509, −7.86157264842483511298107402869, −5.70941848086089266445339497339, −4.50891841420106988357796324547, −3.39480704296945405252045678404, −1.53090805213268930119725659074, 1.60987450182671263655638784436, 4.29972863374502112967357765166, 5.18600007951943416347800487239, 6.74762410876947136090532186280, 7.34284845134376802528979596671, 8.388878055674106088832302406700, 9.415592411392235081344331786611, 10.43166988118358398887893174860, 11.68944938770966772697940139888, 12.74952834532515851328928491658

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