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Properties

Label 2-200-40.27-c1-0-13
Degree $2$
Conductor $200$
Sign $0.229 + 0.973i$
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 − i)2-s − 2i·4-s + (2 − 2i)7-s + (−2 − 2i)8-s + 3i·9-s + 2·11-s + (−4 − 4i)13-s − 4i·14-s − 4·16-s + (3 + 3i)18-s + 6i·19-s + (2 − 2i)22-s + (6 + 6i)23-s − 8·26-s + (−4 − 4i)28-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s − i·4-s + (0.755 − 0.755i)7-s + (−0.707 − 0.707i)8-s + i·9-s + 0.603·11-s + (−1.10 − 1.10i)13-s − 1.06i·14-s − 16-s + (0.707 + 0.707i)18-s + 1.37i·19-s + (0.426 − 0.426i)22-s + (1.25 + 1.25i)23-s − 1.56·26-s + (−0.755 − 0.755i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $0.229 + 0.973i$
Analytic conductor: \(1.59700\)
Root analytic conductor: \(1.26372\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{200} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 200,\ (\ :1/2),\ 0.229 + 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34588 - 1.06515i\)
\(L(\frac12)\) \(\approx\) \(1.34588 - 1.06515i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + i)T \)
5 \( 1 \)
good3 \( 1 - 3iT^{2} \)
7 \( 1 + (-2 + 2i)T - 7iT^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + (4 + 4i)T + 13iT^{2} \)
17 \( 1 + 17iT^{2} \)
19 \( 1 - 6iT - 19T^{2} \)
23 \( 1 + (-6 - 6i)T + 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + (8 - 8i)T - 37iT^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + (-2 + 2i)T - 47iT^{2} \)
53 \( 1 + (4 + 4i)T + 53iT^{2} \)
59 \( 1 + 14iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 + 14iT - 89T^{2} \)
97 \( 1 + 97iT^{2} \)
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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

−12.26343214044480564145329677017, −11.28395701453297447929673370677, −10.49501904596233125654467997024, −9.723013269481225824242636840883, −8.129716278031306848507660235810, −7.12512425759943109273312593627, −5.48631819650078502909857149801, −4.70715514918353254370542366371, −3.31387394047788956235381356104, −1.61606390462301028453704667568, 2.58408328013836953007760711775, 4.25156380407754398270779325137, 5.18572397500845188810684699799, 6.54540621454981498802513765360, 7.23781402617693266289741796492, 8.837796199288290520604214729907, 9.157962624803406978300336661028, 11.12039639515583195070483304497, 12.01058425573912472670383421813, 12.51050852512989233350777280494

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