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Properties

Label 2-20e2-80.67-c1-0-9
Degree $2$
Conductor $400$
Sign $0.582 - 0.813i$
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  + (−0.687 + 1.23i)2-s + 0.614i·3-s + (−1.05 − 1.69i)4-s + (−0.759 − 0.422i)6-s + (−2.83 − 2.83i)7-s + (2.82 − 0.134i)8-s + 2.62·9-s + (1.95 + 1.95i)11-s + (1.04 − 0.647i)12-s + 2.05·13-s + (5.45 − 1.55i)14-s + (−1.77 + 3.58i)16-s + (4.06 + 4.06i)17-s + (−1.80 + 3.24i)18-s + (0.683 + 0.683i)19-s + ⋯
L(s)  = 1  + (−0.486 + 0.873i)2-s + 0.354i·3-s + (−0.527 − 0.849i)4-s + (−0.310 − 0.172i)6-s + (−1.07 − 1.07i)7-s + (0.998 − 0.0473i)8-s + 0.874·9-s + (0.590 + 0.590i)11-s + (0.301 − 0.187i)12-s + 0.569·13-s + (1.45 − 0.415i)14-s + (−0.444 + 0.895i)16-s + (0.986 + 0.986i)17-s + (−0.425 + 0.763i)18-s + (0.156 + 0.156i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.582 - 0.813i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.582 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.582 - 0.813i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.582 - 0.813i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.935288 + 0.480713i\)
\(L(\frac12)\) \(\approx\) \(0.935288 + 0.480713i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.687 - 1.23i)T \)
5 \( 1 \)
good3 \( 1 - 0.614iT - 3T^{2} \)
7 \( 1 + (2.83 + 2.83i)T + 7iT^{2} \)
11 \( 1 + (-1.95 - 1.95i)T + 11iT^{2} \)
13 \( 1 - 2.05T + 13T^{2} \)
17 \( 1 + (-4.06 - 4.06i)T + 17iT^{2} \)
19 \( 1 + (-0.683 - 0.683i)T + 19iT^{2} \)
23 \( 1 + (-4.95 + 4.95i)T - 23iT^{2} \)
29 \( 1 + (0.835 - 0.835i)T - 29iT^{2} \)
31 \( 1 + 2.35iT - 31T^{2} \)
37 \( 1 - 4.54T + 37T^{2} \)
41 \( 1 + 5.07iT - 41T^{2} \)
43 \( 1 - 0.849T + 43T^{2} \)
47 \( 1 + (2.72 - 2.72i)T - 47iT^{2} \)
53 \( 1 - 5.17iT - 53T^{2} \)
59 \( 1 + (-4.16 + 4.16i)T - 59iT^{2} \)
61 \( 1 + (-5.55 - 5.55i)T + 61iT^{2} \)
67 \( 1 - 1.73T + 67T^{2} \)
71 \( 1 - 2.33T + 71T^{2} \)
73 \( 1 + (4.39 + 4.39i)T + 73iT^{2} \)
79 \( 1 + 14.0T + 79T^{2} \)
83 \( 1 + 2.75iT - 83T^{2} \)
89 \( 1 + 11.6T + 89T^{2} \)
97 \( 1 + (-3.52 - 3.52i)T + 97iT^{2} \)
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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

−10.95044642893954203118857434988, −10.14866495742380087668798735619, −9.728930392825462198476874806803, −8.702278335873933890041591453857, −7.47402954950276369728588024113, −6.84938915166797702170207310786, −5.95454310839765982546800432139, −4.48617102454201357536430054219, −3.70795433228154740403655748023, −1.16326077930594474252787873808, 1.16786759150477225777941443752, 2.79703330864710634417221380718, 3.68353101766668319033116447104, 5.27832115199714531899324633400, 6.54123316808848839376539822817, 7.52273567086135370344918400251, 8.686909538431323975114803335528, 9.451886750329245135539685777266, 10.00414850629063984664960613485, 11.37376636047580446962138089046

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