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Properties

Label 2-20e2-80.43-c1-0-18
Degree $2$
Conductor $400$
Sign $0.697 - 0.717i$
Analytic cond. $3.19401$
Root an. cond. $1.78718$
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  + (1.14 + 0.828i)2-s − 0.692i·3-s + (0.627 + 1.89i)4-s + (0.573 − 0.794i)6-s + (0.343 − 0.343i)7-s + (−0.853 + 2.69i)8-s + 2.52·9-s + (0.843 − 0.843i)11-s + (1.31 − 0.434i)12-s + 3.68·13-s + (0.678 − 0.109i)14-s + (−3.21 + 2.38i)16-s + (−0.412 + 0.412i)17-s + (2.88 + 2.08i)18-s + (−5.37 + 5.37i)19-s + ⋯
L(s)  = 1  + (0.810 + 0.585i)2-s − 0.399i·3-s + (0.313 + 0.949i)4-s + (0.234 − 0.324i)6-s + (0.129 − 0.129i)7-s + (−0.301 + 0.953i)8-s + 0.840·9-s + (0.254 − 0.254i)11-s + (0.379 − 0.125i)12-s + 1.02·13-s + (0.181 − 0.0292i)14-s + (−0.802 + 0.596i)16-s + (−0.0999 + 0.0999i)17-s + (0.680 + 0.492i)18-s + (−1.23 + 1.23i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.697 - 0.717i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.697 - 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.697 - 0.717i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.697 - 0.717i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.08800 + 0.882280i\)
\(L(\frac12)\) \(\approx\) \(2.08800 + 0.882280i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.14 - 0.828i)T \)
5 \( 1 \)
good3 \( 1 + 0.692iT - 3T^{2} \)
7 \( 1 + (-0.343 + 0.343i)T - 7iT^{2} \)
11 \( 1 + (-0.843 + 0.843i)T - 11iT^{2} \)
13 \( 1 - 3.68T + 13T^{2} \)
17 \( 1 + (0.412 - 0.412i)T - 17iT^{2} \)
19 \( 1 + (5.37 - 5.37i)T - 19iT^{2} \)
23 \( 1 + (-3.08 - 3.08i)T + 23iT^{2} \)
29 \( 1 + (4.22 + 4.22i)T + 29iT^{2} \)
31 \( 1 + 8.75iT - 31T^{2} \)
37 \( 1 - 5.41T + 37T^{2} \)
41 \( 1 + 2.54iT - 41T^{2} \)
43 \( 1 + 4.30T + 43T^{2} \)
47 \( 1 + (4.56 + 4.56i)T + 47iT^{2} \)
53 \( 1 + 6.07iT - 53T^{2} \)
59 \( 1 + (7.33 + 7.33i)T + 59iT^{2} \)
61 \( 1 + (4.81 - 4.81i)T - 61iT^{2} \)
67 \( 1 + 14.3T + 67T^{2} \)
71 \( 1 + 2.97T + 71T^{2} \)
73 \( 1 + (-6.87 + 6.87i)T - 73iT^{2} \)
79 \( 1 - 10.1T + 79T^{2} \)
83 \( 1 - 7.15iT - 83T^{2} \)
89 \( 1 - 1.10T + 89T^{2} \)
97 \( 1 + (7.15 - 7.15i)T - 97iT^{2} \)
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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

−11.55449713437351075213646258809, −10.75181215515465241266024934632, −9.439403528655293645670432651592, −8.238302908411195039044024368586, −7.61259534689029721072712882917, −6.46702354514011624934411617481, −5.87390017324263360262196023889, −4.40302375629971769227412842078, −3.62642719331490457294038521952, −1.86317585544169625288127150032, 1.52638577605510697770055692955, 3.07526202527956961194773940195, 4.27687578344479294420174243021, 4.94791511119656416017461162951, 6.31705533561526512040458184591, 7.06617767371384588563319309122, 8.720986923120419194974922352030, 9.469879976754019995299173021316, 10.71053099227269201836818290038, 10.90898672925254704273442584197

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