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Properties

Label 2-20e2-400.189-c1-0-12
Degree $2$
Conductor $400$
Sign $-0.675 - 0.737i$
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  + (0.970 + 1.02i)2-s + (−0.133 − 0.262i)3-s + (−0.117 + 1.99i)4-s + (−0.662 + 2.13i)5-s + (0.140 − 0.392i)6-s − 0.849·7-s + (−2.16 + 1.81i)8-s + (1.71 − 2.35i)9-s + (−2.84 + 1.39i)10-s + (−0.801 + 5.06i)11-s + (0.540 − 0.236i)12-s + (0.623 − 0.0986i)13-s + (−0.824 − 0.874i)14-s + (0.650 − 0.111i)15-s + (−3.97 − 0.468i)16-s + (−2.00 + 0.652i)17-s + ⋯
L(s)  = 1  + (0.686 + 0.727i)2-s + (−0.0773 − 0.151i)3-s + (−0.0587 + 0.998i)4-s + (−0.296 + 0.955i)5-s + (0.0573 − 0.160i)6-s − 0.321·7-s + (−0.766 + 0.642i)8-s + (0.570 − 0.785i)9-s + (−0.898 + 0.439i)10-s + (−0.241 + 1.52i)11-s + (0.156 − 0.0682i)12-s + (0.172 − 0.0273i)13-s + (−0.220 − 0.233i)14-s + (0.167 − 0.0288i)15-s + (−0.993 − 0.117i)16-s + (−0.487 + 0.158i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.675 - 0.737i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.675 - 0.737i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ -0.675 - 0.737i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.648445 + 1.47389i\)
\(L(\frac12)\) \(\approx\) \(0.648445 + 1.47389i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.970 - 1.02i)T \)
5 \( 1 + (0.662 - 2.13i)T \)
good3 \( 1 + (0.133 + 0.262i)T + (-1.76 + 2.42i)T^{2} \)
7 \( 1 + 0.849T + 7T^{2} \)
11 \( 1 + (0.801 - 5.06i)T + (-10.4 - 3.39i)T^{2} \)
13 \( 1 + (-0.623 + 0.0986i)T + (12.3 - 4.01i)T^{2} \)
17 \( 1 + (2.00 - 0.652i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (2.34 - 4.60i)T + (-11.1 - 15.3i)T^{2} \)
23 \( 1 + (-5.46 + 3.96i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-7.06 + 3.59i)T + (17.0 - 23.4i)T^{2} \)
31 \( 1 + (-1.85 - 5.71i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (0.174 + 1.10i)T + (-35.1 + 11.4i)T^{2} \)
41 \( 1 + (-0.885 + 1.21i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + (-2.53 - 2.53i)T + 43iT^{2} \)
47 \( 1 + (-9.68 - 3.14i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-8.36 + 4.26i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (-4.87 + 0.772i)T + (56.1 - 18.2i)T^{2} \)
61 \( 1 + (-10.7 - 1.69i)T + (58.0 + 18.8i)T^{2} \)
67 \( 1 + (7.56 + 3.85i)T + (39.3 + 54.2i)T^{2} \)
71 \( 1 + (12.0 + 3.91i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-11.8 + 8.63i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (0.368 - 1.13i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (7.38 + 3.76i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (-9.69 - 13.3i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (2.26 + 0.735i)T + (78.4 + 57.0i)T^{2} \)
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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

−12.03247216766762624933909247821, −10.69487476300842312136169057227, −9.864248913716037546585224450431, −8.635957867829542699841546345417, −7.50031168455665875246047750305, −6.79570779264851885730324885431, −6.23220105523556331243577784305, −4.67031288193172921645674874724, −3.80322544087165679796884444702, −2.51284107323265287622129945978, 0.895727712455194597855547754390, 2.67805595666675188979739595647, 3.98911272562730033066122313173, 4.92082089482363017975254879962, 5.74323981795636401630621556109, 7.02822592279962017011023123242, 8.486368918826400691221913740059, 9.142973980484432700413082752113, 10.29800995270821728637780524271, 11.11546290122344201356370337191

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