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Properties

Label 2-20e2-400.141-c1-0-37
Degree $2$
Conductor $400$
Sign $0.896 + 0.443i$
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 + 0.906i)2-s + (0.313 − 0.615i)3-s + (0.357 − 1.96i)4-s + (1.06 − 1.96i)5-s + (0.217 + 0.951i)6-s + 0.175i·7-s + (1.39 + 2.46i)8-s + (1.48 + 2.04i)9-s + (0.621 + 3.10i)10-s + (−3.06 + 0.485i)11-s + (−1.09 − 0.836i)12-s + (3.42 + 0.541i)13-s + (−0.158 − 0.190i)14-s + (−0.873 − 1.27i)15-s + (−3.74 − 1.40i)16-s + (2.44 − 7.53i)17-s + ⋯
L(s)  = 1  + (−0.767 + 0.640i)2-s + (0.180 − 0.355i)3-s + (0.178 − 0.983i)4-s + (0.477 − 0.878i)5-s + (0.0886 + 0.388i)6-s + 0.0661i·7-s + (0.493 + 0.869i)8-s + (0.494 + 0.680i)9-s + (0.196 + 0.980i)10-s + (−0.924 + 0.146i)11-s + (−0.316 − 0.241i)12-s + (0.949 + 0.150i)13-s + (−0.0423 − 0.0507i)14-s + (−0.225 − 0.328i)15-s + (−0.936 − 0.351i)16-s + (0.593 − 1.82i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 + 0.443i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.896 + 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.896 + 0.443i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.896 + 0.443i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08333 - 0.253474i\)
\(L(\frac12)\) \(\approx\) \(1.08333 - 0.253474i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.08 - 0.906i)T \)
5 \( 1 + (-1.06 + 1.96i)T \)
good3 \( 1 + (-0.313 + 0.615i)T + (-1.76 - 2.42i)T^{2} \)
7 \( 1 - 0.175iT - 7T^{2} \)
11 \( 1 + (3.06 - 0.485i)T + (10.4 - 3.39i)T^{2} \)
13 \( 1 + (-3.42 - 0.541i)T + (12.3 + 4.01i)T^{2} \)
17 \( 1 + (-2.44 + 7.53i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.836 + 0.426i)T + (11.1 - 15.3i)T^{2} \)
23 \( 1 + (-4.13 + 5.69i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (1.76 - 3.46i)T + (-17.0 - 23.4i)T^{2} \)
31 \( 1 + (-0.114 + 0.353i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-0.0815 + 0.514i)T + (-35.1 - 11.4i)T^{2} \)
41 \( 1 + (2.94 + 4.05i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + (7.74 - 7.74i)T - 43iT^{2} \)
47 \( 1 + (0.182 + 0.561i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-4.40 - 2.24i)T + (31.1 + 42.8i)T^{2} \)
59 \( 1 + (-1.29 + 8.17i)T + (-56.1 - 18.2i)T^{2} \)
61 \( 1 + (-1.34 - 8.48i)T + (-58.0 + 18.8i)T^{2} \)
67 \( 1 + (2.18 - 1.11i)T + (39.3 - 54.2i)T^{2} \)
71 \( 1 + (12.3 - 4.02i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-7.18 + 9.88i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-5.06 - 15.5i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-6.39 + 3.25i)T + (48.7 - 67.1i)T^{2} \)
89 \( 1 + (-2.03 + 2.80i)T + (-27.5 - 84.6i)T^{2} \)
97 \( 1 + (-2.15 - 6.63i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.91393017779932403356157403226, −10.11716669959666731197238187497, −9.200317797007350347688106255687, −8.441084069810219883718290911755, −7.58913114829862877137204618577, −6.71525410252617627350608730147, −5.38565964838986248981737294021, −4.82379571424420786007392921222, −2.46994622217632985601691398129, −1.05419038827922466713601174551, 1.59619275537907824082407710786, 3.16567607166626444834364242121, 3.83232772942567615772770840259, 5.71932930539718794689098317793, 6.80673997590219788939123417515, 7.82756079938656221613969451581, 8.745664470611918753914008066835, 9.785148547803993992562095639483, 10.37680557346967396114350793189, 10.97086599098873283984806900400

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