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Properties

Label 2-200-40.27-c1-0-12
Degree $2$
Conductor $200$
Sign $0.973 - 0.229i$
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 + (2 − 2i)3-s + 2i·4-s + 4·6-s + (−2 + 2i)8-s − 5i·9-s − 6·11-s + (4 + 4i)12-s − 4·16-s + (4 + 4i)17-s + (5 − 5i)18-s − 2i·19-s + (−6 − 6i)22-s + 8i·24-s + (−4 − 4i)27-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + (1.15 − 1.15i)3-s + i·4-s + 1.63·6-s + (−0.707 + 0.707i)8-s − 1.66i·9-s − 1.80·11-s + (1.15 + 1.15i)12-s − 16-s + (0.970 + 0.970i)17-s + (1.17 − 1.17i)18-s − 0.458i·19-s + (−1.27 − 1.27i)22-s + 1.63i·24-s + (−0.769 − 0.769i)27-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $0.973 - 0.229i$
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: \((2,\ 200,\ (\ :1/2),\ 0.973 - 0.229i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.11645 + 0.246426i\)
\(L(\frac12)\) \(\approx\) \(2.11645 + 0.246426i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 + 2i)T - 3iT^{2} \)
7 \( 1 - 7iT^{2} \)
11 \( 1 + 6T + 11T^{2} \)
13 \( 1 + 13iT^{2} \)
17 \( 1 + (-4 - 4i)T + 17iT^{2} \)
19 \( 1 + 2iT - 19T^{2} \)
23 \( 1 + 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 37iT^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + (-6 + 6i)T - 43iT^{2} \)
47 \( 1 - 47iT^{2} \)
53 \( 1 + 53iT^{2} \)
59 \( 1 + 6iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + (6 + 6i)T + 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-12 + 12i)T - 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (-2 + 2i)T - 83iT^{2} \)
89 \( 1 - 18iT - 89T^{2} \)
97 \( 1 + (-12 - 12i)T + 97iT^{2} \)
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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

−12.83461125882297384840720534982, −12.16709337580430776584670510375, −10.59309616654074349556559346898, −9.024174749586312479583394624138, −7.940817487311768164169907409616, −7.67781266462110170830955028344, −6.43392652598721456379763884018, −5.20188861543177951230871158341, −3.44399200216977935047946699399, −2.36282794789362496136437965985, 2.53944393185318725803307799768, 3.38425863832263344277539571706, 4.67995772602138674958764101542, 5.53793135770770986041082780606, 7.56544907154702072696127104846, 8.684520954755740637860495321363, 9.906132805346180439449932564624, 10.20659584223364561574361795293, 11.31279950412465471742841070728, 12.59632793749163586086107653210

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