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Properties

Label 2-20e2-100.23-c1-0-1
Degree $2$
Conductor $400$
Sign $-0.762 - 0.646i$
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 + 2.69i)3-s + (−0.681 + 2.12i)5-s + (−2.75 − 2.75i)7-s + (−3.62 + 4.98i)9-s + (−0.220 − 0.303i)11-s + (0.617 + 3.89i)13-s + (−6.67 + 1.08i)15-s + (2.98 + 1.52i)17-s + (1.38 + 4.27i)19-s + (3.64 − 11.2i)21-s + (0.779 − 4.92i)23-s + (−4.07 − 2.90i)25-s + (−9.44 − 1.49i)27-s + (−0.329 − 0.107i)29-s + (−6.71 + 2.18i)31-s + ⋯
L(s)  = 1  + (0.793 + 1.55i)3-s + (−0.304 + 0.952i)5-s + (−1.04 − 1.04i)7-s + (−1.20 + 1.66i)9-s + (−0.0665 − 0.0915i)11-s + (0.171 + 1.08i)13-s + (−1.72 + 0.280i)15-s + (0.724 + 0.369i)17-s + (0.318 + 0.979i)19-s + (0.794 − 2.44i)21-s + (0.162 − 1.02i)23-s + (−0.814 − 0.580i)25-s + (−1.81 − 0.287i)27-s + (−0.0611 − 0.0198i)29-s + (−1.20 + 0.391i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.762 - 0.646i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.762 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.762 - 0.646i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ -0.762 - 0.646i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.472290 + 1.28711i\)
\(L(\frac12)\) \(\approx\) \(0.472290 + 1.28711i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.681 - 2.12i)T \)
good3 \( 1 + (-1.37 - 2.69i)T + (-1.76 + 2.42i)T^{2} \)
7 \( 1 + (2.75 + 2.75i)T + 7iT^{2} \)
11 \( 1 + (0.220 + 0.303i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.617 - 3.89i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (-2.98 - 1.52i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (-1.38 - 4.27i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-0.779 + 4.92i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (0.329 + 0.107i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (6.71 - 2.18i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (-9.58 + 1.51i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (-6.22 - 4.52i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-2.87 + 2.87i)T - 43iT^{2} \)
47 \( 1 + (-10.0 + 5.13i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (3.53 - 1.80i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (-1.67 - 1.21i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (8.21 - 5.97i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (2.13 - 4.18i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (-9.43 - 3.06i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-15.7 - 2.48i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (1.52 - 4.70i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (4.16 + 2.12i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (5.21 + 7.18i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (-2.92 - 5.74i)T + (-57.0 + 78.4i)T^{2} \)
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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

−11.12872803992036490136876184610, −10.54303025208545347212277684616, −9.878154111364195951440978478530, −9.207382472171100893408411138200, −8.007338070935277523058895432264, −7.04480945293377933940215415560, −5.88307740615003623236272655598, −4.16585384223816165258211527616, −3.78524311661992329826819501993, −2.75675792067351867665812847854, 0.837751856365797975462852762400, 2.46306179957559199456677312532, 3.40824089175215100681535072130, 5.39606823215241320508160232053, 6.17930031298661422048185530099, 7.50263010023965805513206173472, 7.935820906914381983471342244769, 9.228934401585259521543829173208, 9.315112035709256388178278644455, 11.27558863170876924749828080637

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