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Properties

Label 2-200-40.29-c1-0-8
Degree $2$
Conductor $200$
Sign $0.316 - 0.948i$
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 + 2.73·3-s + (−1.73 + i)4-s + (1 + 3.73i)6-s − 0.732i·7-s + (−2 − 1.99i)8-s + 4.46·9-s + 2i·11-s + (−4.73 + 2.73i)12-s − 3.46·13-s + (1 − 0.267i)14-s + (1.99 − 3.46i)16-s − 3.46i·17-s + (1.63 + 6.09i)18-s + 0.535i·19-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s + 1.57·3-s + (−0.866 + 0.5i)4-s + (0.408 + 1.52i)6-s − 0.276i·7-s + (−0.707 − 0.707i)8-s + 1.48·9-s + 0.603i·11-s + (−1.36 + 0.788i)12-s − 0.960·13-s + (0.267 − 0.0716i)14-s + (0.499 − 0.866i)16-s − 0.840i·17-s + (0.385 + 1.43i)18-s + 0.122i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $0.316 - 0.948i$
Analytic conductor: \(1.59700\)
Root analytic conductor: \(1.26372\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{200} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 200,\ (\ :1/2),\ 0.316 - 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51596 + 1.09264i\)
\(L(\frac12)\) \(\approx\) \(1.51596 + 1.09264i\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.73T + 3T^{2} \)
7 \( 1 + 0.732iT - 7T^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
13 \( 1 + 3.46T + 13T^{2} \)
17 \( 1 + 3.46iT - 17T^{2} \)
19 \( 1 - 0.535iT - 19T^{2} \)
23 \( 1 + 6.19iT - 23T^{2} \)
29 \( 1 - 6.92iT - 29T^{2} \)
31 \( 1 + 5.46T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 - 1.46T + 41T^{2} \)
43 \( 1 + 5.26T + 43T^{2} \)
47 \( 1 + 3.26iT - 47T^{2} \)
53 \( 1 - 11.4T + 53T^{2} \)
59 \( 1 + 7.46iT - 59T^{2} \)
61 \( 1 - 8.92iT - 61T^{2} \)
67 \( 1 + 10.7T + 67T^{2} \)
71 \( 1 - 5.46T + 71T^{2} \)
73 \( 1 - 7.46iT - 73T^{2} \)
79 \( 1 - 1.07T + 79T^{2} \)
83 \( 1 - 1.26T + 83T^{2} \)
89 \( 1 + 8.92T + 89T^{2} \)
97 \( 1 - 14.3iT - 97T^{2} \)
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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

−13.00978068244053554909483701038, −12.16730480470387234062038022272, −10.23167650073203254228294314385, −9.317267240410379258407500104120, −8.574460175233462015115753246520, −7.50281848193954513407267736321, −6.93615623970837667152620247320, −5.08516966199271487860207398171, −3.95361207165823130152554861631, −2.62640780843944343045930744331, 2.00285850582407157288296123012, 3.11540259944749621306411411312, 4.15046156057961027481250194208, 5.69285397887016362610639577947, 7.56978981097575224447493628148, 8.545406457889532750429173903124, 9.346053563369005924852078355367, 10.11645190793984664776363213561, 11.35943792220223732568424433903, 12.41380677571119989898439094718

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