Properties

Label 2-20e2-400.181-c1-0-15
Degree $2$
Conductor $400$
Sign $0.878 - 0.477i$
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  + (−1.37 + 0.340i)2-s + (0.176 − 1.11i)3-s + (1.76 − 0.933i)4-s + (0.507 − 2.17i)5-s + (0.136 + 1.59i)6-s + 4.59i·7-s + (−2.11 + 1.88i)8-s + (1.63 + 0.532i)9-s + (0.0439 + 3.16i)10-s + (−2.56 + 5.03i)11-s + (−0.728 − 2.13i)12-s + (1.56 + 3.07i)13-s + (−1.56 − 6.30i)14-s + (−2.33 − 0.950i)15-s + (2.25 − 3.30i)16-s + (0.981 + 0.713i)17-s + ⋯
L(s)  = 1  + (−0.970 + 0.240i)2-s + (0.102 − 0.644i)3-s + (0.884 − 0.466i)4-s + (0.226 − 0.973i)5-s + (0.0558 + 0.649i)6-s + 1.73i·7-s + (−0.746 + 0.665i)8-s + (0.546 + 0.177i)9-s + (0.0138 + 0.999i)10-s + (−0.773 + 1.51i)11-s + (−0.210 − 0.617i)12-s + (0.434 + 0.852i)13-s + (−0.417 − 1.68i)14-s + (−0.604 − 0.245i)15-s + (0.564 − 0.825i)16-s + (0.238 + 0.173i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.878 - 0.477i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.878 - 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.878 - 0.477i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.878 - 0.477i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.949538 + 0.241283i\)
\(L(\frac12)\) \(\approx\) \(0.949538 + 0.241283i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.37 - 0.340i)T \)
5 \( 1 + (-0.507 + 2.17i)T \)
good3 \( 1 + (-0.176 + 1.11i)T + (-2.85 - 0.927i)T^{2} \)
7 \( 1 - 4.59iT - 7T^{2} \)
11 \( 1 + (2.56 - 5.03i)T + (-6.46 - 8.89i)T^{2} \)
13 \( 1 + (-1.56 - 3.07i)T + (-7.64 + 10.5i)T^{2} \)
17 \( 1 + (-0.981 - 0.713i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-6.70 + 1.06i)T + (18.0 - 5.87i)T^{2} \)
23 \( 1 + (1.41 - 0.459i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (-1.08 + 6.83i)T + (-27.5 - 8.96i)T^{2} \)
31 \( 1 + (-1.40 - 1.01i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (4.04 - 2.06i)T + (21.7 - 29.9i)T^{2} \)
41 \( 1 + (1.56 + 0.507i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (-1.39 - 1.39i)T + 43iT^{2} \)
47 \( 1 + (6.49 - 4.72i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-3.66 - 0.580i)T + (50.4 + 16.3i)T^{2} \)
59 \( 1 + (-12.0 + 6.12i)T + (34.6 - 47.7i)T^{2} \)
61 \( 1 + (-1.53 - 0.782i)T + (35.8 + 49.3i)T^{2} \)
67 \( 1 + (-1.81 + 0.287i)T + (63.7 - 20.7i)T^{2} \)
71 \( 1 + (2.40 + 3.31i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-8.13 + 2.64i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (9.12 - 6.62i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (3.18 - 0.505i)T + (78.9 - 25.6i)T^{2} \)
89 \( 1 + (4.23 - 1.37i)T + (72.0 - 52.3i)T^{2} \)
97 \( 1 + (-3.31 + 2.40i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.59836145315416785915743764160, −9.861125461922150862977206334302, −9.594397870358144025821465051659, −8.517356011755757429651764853959, −7.85189594534713326581063534829, −6.83384464273279427603021239973, −5.72097293891843048254286808159, −4.84875308108342982833677395166, −2.38912436933311429200351253833, −1.60079941991860182909438125624, 0.958430774024420732333974418882, 3.21470391276757278112618149789, 3.60973588113733925367977130914, 5.55053522503547724400573658362, 6.87764664292795183250908064699, 7.50618915731938775688269805148, 8.440524696804810675003886423394, 9.800646948590920720458940937915, 10.33944604195197372975781782036, 10.74115273720790740300604417255

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