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Properties

Label 2-20e2-100.79-c0-0-0
Degree $2$
Conductor $400$
Sign $0.929 + 0.368i$
Analytic cond. $0.199626$
Root an. cond. $0.446795$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 − 0.587i)5-s + (−0.309 − 0.951i)9-s + (−1.11 + 0.363i)13-s + (1.11 + 1.53i)17-s + (0.309 − 0.951i)25-s + (−0.5 − 0.363i)29-s + (−1.80 + 0.587i)37-s + (0.5 + 1.53i)41-s + (−0.809 − 0.587i)45-s − 49-s + (−0.690 + 0.951i)53-s + (0.5 − 1.53i)61-s + (−0.690 + 0.951i)65-s + (−1.11 − 0.363i)73-s + (−0.809 + 0.587i)81-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)5-s + (−0.309 − 0.951i)9-s + (−1.11 + 0.363i)13-s + (1.11 + 1.53i)17-s + (0.309 − 0.951i)25-s + (−0.5 − 0.363i)29-s + (−1.80 + 0.587i)37-s + (0.5 + 1.53i)41-s + (−0.809 − 0.587i)45-s − 49-s + (−0.690 + 0.951i)53-s + (0.5 − 1.53i)61-s + (−0.690 + 0.951i)65-s + (−1.11 − 0.363i)73-s + (−0.809 + 0.587i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.929 + 0.368i$
Analytic conductor: \(0.199626\)
Root analytic conductor: \(0.446795\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :0),\ 0.929 + 0.368i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8987725254\)
\(L(\frac12)\) \(\approx\) \(0.8987725254\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.809 + 0.587i)T \)
good3 \( 1 + (0.309 + 0.951i)T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (1.11 - 0.363i)T + (0.809 - 0.587i)T^{2} \)
17 \( 1 + (-1.11 - 1.53i)T + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + (-0.809 - 0.587i)T^{2} \)
29 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (1.80 - 0.587i)T + (0.809 - 0.587i)T^{2} \)
41 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.690 - 0.951i)T + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
67 \( 1 + (0.309 - 0.951i)T^{2} \)
71 \( 1 + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (1.11 + 0.363i)T + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (0.309 - 0.951i)T^{2} \)
89 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
97 \( 1 + (-1.11 + 1.53i)T + (-0.309 - 0.951i)T^{2} \)
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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

−11.60109897579176044144238305398, −10.26448278078356289299910946481, −9.676549757626723142209792648681, −8.816837374992362635113816311346, −7.84414524235281552958243320580, −6.52427110502955273373945286963, −5.75806364247508604987487712696, −4.65983919067586011978452406047, −3.28986863138439458539880391728, −1.67747576978274899193502685885, 2.16399591947852327969843929509, 3.18756660000288309152276272385, 5.06067267484679542299630247484, 5.56969426992634278410032912524, 7.05623045946663616135209441725, 7.60550967556955935521421202044, 8.952153147393279456436444436742, 9.895876285656303192517067452653, 10.49291110913306791471921002321, 11.47447693944911404871018336081

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