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Properties

Label 2-20e2-80.3-c1-0-4
Degree $2$
Conductor $400$
Sign $-0.825 - 0.564i$
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  + (−1.08 + 0.911i)2-s + 0.619·3-s + (0.337 − 1.97i)4-s + (−0.669 + 0.564i)6-s + (−1.82 + 1.82i)7-s + (1.43 + 2.43i)8-s − 2.61·9-s + (−0.567 − 0.567i)11-s + (0.208 − 1.22i)12-s + 2.78i·13-s + (0.308 − 3.63i)14-s + (−3.77 − 1.32i)16-s + (−3.65 + 3.65i)17-s + (2.82 − 2.38i)18-s + (4.51 + 4.51i)19-s + ⋯
L(s)  = 1  + (−0.764 + 0.644i)2-s + 0.357·3-s + (0.168 − 0.985i)4-s + (−0.273 + 0.230i)6-s + (−0.689 + 0.689i)7-s + (0.506 + 0.862i)8-s − 0.872·9-s + (−0.171 − 0.171i)11-s + (0.0602 − 0.352i)12-s + 0.773i·13-s + (0.0824 − 0.971i)14-s + (−0.943 − 0.332i)16-s + (−0.885 + 0.885i)17-s + (0.666 − 0.562i)18-s + (1.03 + 1.03i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.825 - 0.564i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.825 - 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.825 - 0.564i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ -0.825 - 0.564i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.181539 + 0.586685i\)
\(L(\frac12)\) \(\approx\) \(0.181539 + 0.586685i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.08 - 0.911i)T \)
5 \( 1 \)
good3 \( 1 - 0.619T + 3T^{2} \)
7 \( 1 + (1.82 - 1.82i)T - 7iT^{2} \)
11 \( 1 + (0.567 + 0.567i)T + 11iT^{2} \)
13 \( 1 - 2.78iT - 13T^{2} \)
17 \( 1 + (3.65 - 3.65i)T - 17iT^{2} \)
19 \( 1 + (-4.51 - 4.51i)T + 19iT^{2} \)
23 \( 1 + (-2.15 - 2.15i)T + 23iT^{2} \)
29 \( 1 + (3.20 - 3.20i)T - 29iT^{2} \)
31 \( 1 - 3.54iT - 31T^{2} \)
37 \( 1 - 5.22iT - 37T^{2} \)
41 \( 1 + 8.76iT - 41T^{2} \)
43 \( 1 + 10.8iT - 43T^{2} \)
47 \( 1 + (3.22 + 3.22i)T + 47iT^{2} \)
53 \( 1 + 12.8T + 53T^{2} \)
59 \( 1 + (3.79 - 3.79i)T - 59iT^{2} \)
61 \( 1 + (-6.63 - 6.63i)T + 61iT^{2} \)
67 \( 1 + 7.78iT - 67T^{2} \)
71 \( 1 - 13.6T + 71T^{2} \)
73 \( 1 + (1.34 - 1.34i)T - 73iT^{2} \)
79 \( 1 - 16.3T + 79T^{2} \)
83 \( 1 - 0.391T + 83T^{2} \)
89 \( 1 + 18.0T + 89T^{2} \)
97 \( 1 + (-6.43 + 6.43i)T - 97iT^{2} \)
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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

−11.45930624393083670456894113320, −10.56396579329467825712523475840, −9.460903837663264105513741869485, −8.906933148091299072592838330869, −8.148626815096626697646579137460, −6.99880807139023546712348049066, −6.06205325590259187330082214515, −5.25076415317958334800372926675, −3.43752891210564289644411041532, −1.97468134486905825181986172028, 0.46965235927743115338303901468, 2.58925729852707619934958758936, 3.34856198677549040023467501439, 4.79425169840586349506868894783, 6.43094629936581124591802490899, 7.44883883291759079080065933681, 8.211792688495369208663349305333, 9.412045432165302871799886324757, 9.695414754527833457433658454227, 11.14278768495379313333834222674

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