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Properties

Label 2-20e2-400.123-c1-0-53
Degree $2$
Conductor $400$
Sign $-0.938 + 0.345i$
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  + (0.325 − 1.37i)2-s + (0.591 + 0.192i)3-s + (−1.78 − 0.895i)4-s + (−2.13 + 0.667i)5-s + (0.456 − 0.750i)6-s + (2.61 − 2.61i)7-s + (−1.81 + 2.16i)8-s + (−2.11 − 1.53i)9-s + (0.223 + 3.15i)10-s + (0.494 − 3.11i)11-s + (−0.884 − 0.872i)12-s + (−1.00 − 0.733i)13-s + (−2.74 − 4.44i)14-s + (−1.38 − 0.0154i)15-s + (2.39 + 3.20i)16-s + (−5.14 − 2.62i)17-s + ⋯
L(s)  = 1  + (0.230 − 0.973i)2-s + (0.341 + 0.110i)3-s + (−0.894 − 0.447i)4-s + (−0.954 + 0.298i)5-s + (0.186 − 0.306i)6-s + (0.988 − 0.988i)7-s + (−0.641 + 0.767i)8-s + (−0.704 − 0.512i)9-s + (0.0707 + 0.997i)10-s + (0.148 − 0.940i)11-s + (−0.255 − 0.251i)12-s + (−0.279 − 0.203i)13-s + (−0.734 − 1.18i)14-s + (−0.358 − 0.00400i)15-s + (0.598 + 0.800i)16-s + (−1.24 − 0.636i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.938 + 0.345i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.938 + 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.938 + 0.345i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ -0.938 + 0.345i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.190466 - 1.06852i\)
\(L(\frac12)\) \(\approx\) \(0.190466 - 1.06852i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.325 + 1.37i)T \)
5 \( 1 + (2.13 - 0.667i)T \)
good3 \( 1 + (-0.591 - 0.192i)T + (2.42 + 1.76i)T^{2} \)
7 \( 1 + (-2.61 + 2.61i)T - 7iT^{2} \)
11 \( 1 + (-0.494 + 3.11i)T + (-10.4 - 3.39i)T^{2} \)
13 \( 1 + (1.00 + 0.733i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (5.14 + 2.62i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (-2.65 - 1.35i)T + (11.1 + 15.3i)T^{2} \)
23 \( 1 + (5.17 + 0.818i)T + (21.8 + 7.10i)T^{2} \)
29 \( 1 + (-2.82 + 1.43i)T + (17.0 - 23.4i)T^{2} \)
31 \( 1 + (4.97 - 1.61i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (-5.09 - 3.70i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-2.88 + 3.97i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 - 9.48T + 43T^{2} \)
47 \( 1 + (-4.36 + 2.22i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (-10.4 - 3.38i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (0.493 + 3.11i)T + (-56.1 + 18.2i)T^{2} \)
61 \( 1 + (1.00 - 6.37i)T + (-58.0 - 18.8i)T^{2} \)
67 \( 1 + (1.20 + 3.69i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (3.49 - 10.7i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-1.75 + 11.1i)T + (-69.4 - 22.5i)T^{2} \)
79 \( 1 + (3.24 - 10.0i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (2.30 - 0.748i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (-8.15 + 5.92i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (5.94 + 11.6i)T + (-57.0 + 78.4i)T^{2} \)
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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

−11.13641770208084378408107301442, −10.26418544387414953501200977883, −9.004531866318761644670800331220, −8.306283948294576258020718330496, −7.36762402020341850203948538672, −5.84835672037851513475684401796, −4.46559271479704813879714910125, −3.78297285004314445916047864362, −2.64185912797250112377162441959, −0.64371433989260048946069304926, 2.34876418998856900784248327411, 4.08989724792488110197610440096, 4.88805505359540860307262219244, 5.86739546153660993886486241177, 7.27929833618699952383799184839, 7.926787969151965749770063067273, 8.689255397753081178382338206318, 9.316053598109317757426302509934, 10.99721477907747227555165746132, 11.86580018287466760612530617654

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