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Properties

Label 2-20e2-80.43-c1-0-5
Degree $2$
Conductor $400$
Sign $-0.107 - 0.994i$
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  + (0.558 − 1.29i)2-s + 2.55i·3-s + (−1.37 − 1.45i)4-s + (3.31 + 1.42i)6-s + (−2.40 + 2.40i)7-s + (−2.65 + 0.977i)8-s − 3.51·9-s + (−2.67 + 2.67i)11-s + (3.70 − 3.51i)12-s + 2.40·13-s + (1.78 + 4.46i)14-s + (−0.212 + 3.99i)16-s + (0.0750 − 0.0750i)17-s + (−1.96 + 4.56i)18-s + (−2.67 + 2.67i)19-s + ⋯
L(s)  = 1  + (0.394 − 0.918i)2-s + 1.47i·3-s + (−0.688 − 0.725i)4-s + (1.35 + 0.581i)6-s + (−0.908 + 0.908i)7-s + (−0.938 + 0.345i)8-s − 1.17·9-s + (−0.807 + 0.807i)11-s + (1.06 − 1.01i)12-s + 0.666·13-s + (0.475 + 1.19i)14-s + (−0.0532 + 0.998i)16-s + (0.0182 − 0.0182i)17-s + (−0.462 + 1.07i)18-s + (−0.613 + 0.613i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.107 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.107 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.107 - 0.994i$
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: \((2,\ 400,\ (\ :1/2),\ -0.107 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.669550 + 0.745785i\)
\(L(\frac12)\) \(\approx\) \(0.669550 + 0.745785i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.558 + 1.29i)T \)
5 \( 1 \)
good3 \( 1 - 2.55iT - 3T^{2} \)
7 \( 1 + (2.40 - 2.40i)T - 7iT^{2} \)
11 \( 1 + (2.67 - 2.67i)T - 11iT^{2} \)
13 \( 1 - 2.40T + 13T^{2} \)
17 \( 1 + (-0.0750 + 0.0750i)T - 17iT^{2} \)
19 \( 1 + (2.67 - 2.67i)T - 19iT^{2} \)
23 \( 1 + (2.12 + 2.12i)T + 23iT^{2} \)
29 \( 1 + (-3.95 - 3.95i)T + 29iT^{2} \)
31 \( 1 + 1.65iT - 31T^{2} \)
37 \( 1 - 2.53T + 37T^{2} \)
41 \( 1 - 1.70iT - 41T^{2} \)
43 \( 1 - 3.84T + 43T^{2} \)
47 \( 1 + (-2.15 - 2.15i)T + 47iT^{2} \)
53 \( 1 - 1.29iT - 53T^{2} \)
59 \( 1 + (5.29 + 5.29i)T + 59iT^{2} \)
61 \( 1 + (-10.2 + 10.2i)T - 61iT^{2} \)
67 \( 1 - 10.6T + 67T^{2} \)
71 \( 1 - 2.27T + 71T^{2} \)
73 \( 1 + (9.99 - 9.99i)T - 73iT^{2} \)
79 \( 1 - 8.70T + 79T^{2} \)
83 \( 1 + 11.1iT - 83T^{2} \)
89 \( 1 + 15.6T + 89T^{2} \)
97 \( 1 + (5.00 - 5.00i)T - 97iT^{2} \)
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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

−11.31978087906185641000129671635, −10.47601881198556829048868850687, −9.909026491246717755403441788188, −9.217059481829589473928811733801, −8.335315001655924488144640253095, −6.31757497190841568794490966239, −5.39808639176907510681578744909, −4.46122737261814179406821437143, −3.49005132956118928114105257146, −2.43212818830717142068971307538, 0.56597428228492349738774799569, 2.81141760210618715512168824332, 4.05947578823624324881312179049, 5.68296610182593157384001642525, 6.43952046650238454756966361313, 7.13290104874731193533567066997, 7.965629776165442729792230465701, 8.713843495417944272491076858413, 10.05710736898850431842205397349, 11.24887692659280776220000052389

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