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Properties

Label 2-200-40.29-c5-0-55
Degree $2$
Conductor $200$
Sign $0.877 + 0.478i$
Analytic cond. $32.0767$
Root an. cond. $5.66363$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.75 − 4.22i)2-s + 25.0·3-s + (−3.78 + 31.7i)4-s + (−94.1 − 105. i)6-s − 103. i·7-s + (148. − 103. i)8-s + 384.·9-s + 740. i·11-s + (−94.7 + 796. i)12-s + 892.·13-s + (−438. + 389. i)14-s + (−995. − 240. i)16-s − 1.13e3i·17-s + (−1.44e3 − 1.62e3i)18-s + 1.15e3i·19-s + ⋯
L(s)  = 1  + (−0.664 − 0.747i)2-s + 1.60·3-s + (−0.118 + 0.992i)4-s + (−1.06 − 1.20i)6-s − 0.799i·7-s + (0.820 − 0.571i)8-s + 1.58·9-s + 1.84i·11-s + (−0.189 + 1.59i)12-s + 1.46·13-s + (−0.597 + 0.530i)14-s + (−0.972 − 0.234i)16-s − 0.955i·17-s + (−1.05 − 1.18i)18-s + 0.734i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 + 0.478i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.877 + 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $0.877 + 0.478i$
Analytic conductor: \(32.0767\)
Root analytic conductor: \(5.66363\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{200} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 200,\ (\ :5/2),\ 0.877 + 0.478i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.733457433\)
\(L(\frac12)\) \(\approx\) \(2.733457433\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.75 + 4.22i)T \)
5 \( 1 \)
good3 \( 1 - 25.0T + 243T^{2} \)
7 \( 1 + 103. iT - 1.68e4T^{2} \)
11 \( 1 - 740. iT - 1.61e5T^{2} \)
13 \( 1 - 892.T + 3.71e5T^{2} \)
17 \( 1 + 1.13e3iT - 1.41e6T^{2} \)
19 \( 1 - 1.15e3iT - 2.47e6T^{2} \)
23 \( 1 - 1.60e3iT - 6.43e6T^{2} \)
29 \( 1 + 2.15e3iT - 2.05e7T^{2} \)
31 \( 1 - 4.95e3T + 2.86e7T^{2} \)
37 \( 1 - 4.40e3T + 6.93e7T^{2} \)
41 \( 1 - 3.78e3T + 1.15e8T^{2} \)
43 \( 1 + 1.30e4T + 1.47e8T^{2} \)
47 \( 1 - 8.00e3iT - 2.29e8T^{2} \)
53 \( 1 - 3.43e4T + 4.18e8T^{2} \)
59 \( 1 + 2.20e4iT - 7.14e8T^{2} \)
61 \( 1 - 2.82e3iT - 8.44e8T^{2} \)
67 \( 1 - 5.49e4T + 1.35e9T^{2} \)
71 \( 1 - 4.28e4T + 1.80e9T^{2} \)
73 \( 1 + 2.08e4iT - 2.07e9T^{2} \)
79 \( 1 + 3.02e4T + 3.07e9T^{2} \)
83 \( 1 + 9.39e4T + 3.93e9T^{2} \)
89 \( 1 + 1.17e3T + 5.58e9T^{2} \)
97 \( 1 + 7.41e4iT - 8.58e9T^{2} \)
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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

−11.38507466963049082752801211809, −10.06040779764080395934479630238, −9.667672272185071130212273909565, −8.578294608770426196125314651624, −7.74114271277496879280170273779, −6.98954755075212374936151715629, −4.36740576079605754911539238250, −3.56450683729495510949285530676, −2.30714960413351034597310421570, −1.23548968953120441800403208342, 1.06278345544151081211379090368, 2.56764840532515714929686137937, 3.81772987291375554962085126437, 5.66657848889535128818380000120, 6.62043604608160208352930742141, 8.225749393206241914144000861472, 8.505734170732484514363380865924, 9.077800331427824563861996548547, 10.39296963616183586975303034477, 11.36348172837691930440623195170

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