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Properties

Label 2-20e2-80.27-c1-0-15
Degree $2$
Conductor $400$
Sign $-0.760 + 0.649i$
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  + (−0.567 − 1.29i)2-s − 1.96·3-s + (−1.35 + 1.47i)4-s + (1.11 + 2.54i)6-s + (1.60 + 1.60i)7-s + (2.67 + 0.920i)8-s + 0.851·9-s + (0.754 − 0.754i)11-s + (2.65 − 2.88i)12-s − 5.94i·13-s + (1.16 − 2.98i)14-s + (−0.327 − 3.98i)16-s + (−1.95 − 1.95i)17-s + (−0.483 − 1.10i)18-s + (0.780 − 0.780i)19-s + ⋯
L(s)  = 1  + (−0.401 − 0.915i)2-s − 1.13·3-s + (−0.677 + 0.735i)4-s + (0.454 + 1.03i)6-s + (0.605 + 0.605i)7-s + (0.945 + 0.325i)8-s + 0.283·9-s + (0.227 − 0.227i)11-s + (0.767 − 0.833i)12-s − 1.64i·13-s + (0.311 − 0.797i)14-s + (−0.0817 − 0.996i)16-s + (−0.474 − 0.474i)17-s + (−0.113 − 0.259i)18-s + (0.179 − 0.179i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.760 + 0.649i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.760 + 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.760 + 0.649i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ -0.760 + 0.649i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.182210 - 0.494267i\)
\(L(\frac12)\) \(\approx\) \(0.182210 - 0.494267i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.567 + 1.29i)T \)
5 \( 1 \)
good3 \( 1 + 1.96T + 3T^{2} \)
7 \( 1 + (-1.60 - 1.60i)T + 7iT^{2} \)
11 \( 1 + (-0.754 + 0.754i)T - 11iT^{2} \)
13 \( 1 + 5.94iT - 13T^{2} \)
17 \( 1 + (1.95 + 1.95i)T + 17iT^{2} \)
19 \( 1 + (-0.780 + 0.780i)T - 19iT^{2} \)
23 \( 1 + (4.93 - 4.93i)T - 23iT^{2} \)
29 \( 1 + (1.44 + 1.44i)T + 29iT^{2} \)
31 \( 1 + 3.60iT - 31T^{2} \)
37 \( 1 + 10.2iT - 37T^{2} \)
41 \( 1 + 6.93iT - 41T^{2} \)
43 \( 1 + 9.91iT - 43T^{2} \)
47 \( 1 + (0.104 - 0.104i)T - 47iT^{2} \)
53 \( 1 - 4.03T + 53T^{2} \)
59 \( 1 + (3.46 + 3.46i)T + 59iT^{2} \)
61 \( 1 + (-0.680 + 0.680i)T - 61iT^{2} \)
67 \( 1 - 9.04iT - 67T^{2} \)
71 \( 1 + 3.64T + 71T^{2} \)
73 \( 1 + (-2.94 - 2.94i)T + 73iT^{2} \)
79 \( 1 - 10.7T + 79T^{2} \)
83 \( 1 - 4.23T + 83T^{2} \)
89 \( 1 - 0.0426T + 89T^{2} \)
97 \( 1 + (-1.91 - 1.91i)T + 97iT^{2} \)
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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

−10.99798845598618456269406096796, −10.34182174311131042889385639387, −9.248651990141516586044377052199, −8.303447656256795152307621678686, −7.36810144433944171666856640097, −5.72102549419078952065681249119, −5.22317986640820735974517731393, −3.76914665507446719019692705433, −2.28779342583009964449256909754, −0.48754153989296951952430250048, 1.46337377976962332208232672478, 4.31578181807383942760097085870, 4.86317284348786821723727670298, 6.28277778559404085025760360146, 6.61878108554287235516640618229, 7.82343594808717282601645861084, 8.765497784874570085755250263281, 9.827630165520842304738577570514, 10.71968977425882088969322851093, 11.44021233452730321803642884842

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