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Properties

Label 2-20e2-5.2-c2-0-15
Degree $2$
Conductor $400$
Sign $-0.525 + 0.850i$
Analytic cond. $10.8992$
Root an. cond. $3.30139$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3 − 3i)3-s + (−3 − 3i)7-s − 9i·9-s − 12·11-s + (12 − 12i)13-s + (−12 − 12i)17-s − 20i·19-s − 18·21-s + (3 − 3i)23-s − 30i·29-s + 8·31-s + (−36 + 36i)33-s + (48 + 48i)37-s − 72i·39-s − 48·41-s + ⋯
L(s)  = 1  + (1 − i)3-s + (−0.428 − 0.428i)7-s − i·9-s − 1.09·11-s + (0.923 − 0.923i)13-s + (−0.705 − 0.705i)17-s − 1.05i·19-s − 0.857·21-s + (0.130 − 0.130i)23-s − 1.03i·29-s + 0.258·31-s + (−1.09 + 1.09i)33-s + (1.29 + 1.29i)37-s − 1.84i·39-s − 1.17·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.525 + 0.850i$
Analytic conductor: \(10.8992\)
Root analytic conductor: \(3.30139\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1),\ -0.525 + 0.850i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.933709 - 1.67470i\)
\(L(\frac12)\) \(\approx\) \(0.933709 - 1.67470i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-3 + 3i)T - 9iT^{2} \)
7 \( 1 + (3 + 3i)T + 49iT^{2} \)
11 \( 1 + 12T + 121T^{2} \)
13 \( 1 + (-12 + 12i)T - 169iT^{2} \)
17 \( 1 + (12 + 12i)T + 289iT^{2} \)
19 \( 1 + 20iT - 361T^{2} \)
23 \( 1 + (-3 + 3i)T - 529iT^{2} \)
29 \( 1 + 30iT - 841T^{2} \)
31 \( 1 - 8T + 961T^{2} \)
37 \( 1 + (-48 - 48i)T + 1.36e3iT^{2} \)
41 \( 1 + 48T + 1.68e3T^{2} \)
43 \( 1 + (27 - 27i)T - 1.84e3iT^{2} \)
47 \( 1 + (-27 - 27i)T + 2.20e3iT^{2} \)
53 \( 1 + (-12 + 12i)T - 2.80e3iT^{2} \)
59 \( 1 - 60iT - 3.48e3T^{2} \)
61 \( 1 - 32T + 3.72e3T^{2} \)
67 \( 1 + (3 + 3i)T + 4.48e3iT^{2} \)
71 \( 1 - 48T + 5.04e3T^{2} \)
73 \( 1 + (-12 + 12i)T - 5.32e3iT^{2} \)
79 \( 1 + 40iT - 6.24e3T^{2} \)
83 \( 1 + (-93 + 93i)T - 6.88e3iT^{2} \)
89 \( 1 - 30iT - 7.92e3T^{2} \)
97 \( 1 + (12 + 12i)T + 9.40e3iT^{2} \)
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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

−10.72458175013660918899007493321, −9.762200093610540505237693532766, −8.634734448814526651943708523415, −7.997532272068241714216158431501, −7.15701527832404135082446591706, −6.24667956975343857049687817858, −4.81405399984980270801786351033, −3.22672451318060338373723014491, −2.43870676023969589162237627710, −0.72747279136474027612099661197, 2.13607627564002681267071726745, 3.37308028498856193386153598131, 4.19306810438204231847214840536, 5.47141942814565829961468910398, 6.62232716123270145599430820839, 8.046098344605982634052705512875, 8.720518630164331904751601750397, 9.454610981170812238437921104651, 10.34956344924694387662957205145, 11.05925817285144134888551415893

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