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Properties

Label 2-20e2-80.67-c1-0-12
Degree $2$
Conductor $400$
Sign $0.510 + 0.859i$
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  + (−0.759 − 1.19i)2-s − 1.39i·3-s + (−0.846 + 1.81i)4-s + (−1.66 + 1.05i)6-s + (2.13 + 2.13i)7-s + (2.80 − 0.366i)8-s + 1.05·9-s + (2.17 + 2.17i)11-s + (2.52 + 1.17i)12-s − 1.54·13-s + (0.925 − 4.16i)14-s + (−2.56 − 3.06i)16-s + (3.86 + 3.86i)17-s + (−0.804 − 1.26i)18-s + (−0.0136 − 0.0136i)19-s + ⋯
L(s)  = 1  + (−0.536 − 0.843i)2-s − 0.804i·3-s + (−0.423 + 0.905i)4-s + (−0.678 + 0.431i)6-s + (0.806 + 0.806i)7-s + (0.991 − 0.129i)8-s + 0.353·9-s + (0.654 + 0.654i)11-s + (0.728 + 0.340i)12-s − 0.428·13-s + (0.247 − 1.11i)14-s + (−0.641 − 0.766i)16-s + (0.937 + 0.937i)17-s + (−0.189 − 0.297i)18-s + (−0.00313 − 0.00313i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.510 + 0.859i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.510 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.510 + 0.859i$
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: \((2,\ 400,\ (\ :1/2),\ 0.510 + 0.859i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02863 - 0.585550i\)
\(L(\frac12)\) \(\approx\) \(1.02863 - 0.585550i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.759 + 1.19i)T \)
5 \( 1 \)
good3 \( 1 + 1.39iT - 3T^{2} \)
7 \( 1 + (-2.13 - 2.13i)T + 7iT^{2} \)
11 \( 1 + (-2.17 - 2.17i)T + 11iT^{2} \)
13 \( 1 + 1.54T + 13T^{2} \)
17 \( 1 + (-3.86 - 3.86i)T + 17iT^{2} \)
19 \( 1 + (0.0136 + 0.0136i)T + 19iT^{2} \)
23 \( 1 + (-3.15 + 3.15i)T - 23iT^{2} \)
29 \( 1 + (3.33 - 3.33i)T - 29iT^{2} \)
31 \( 1 + 8.92iT - 31T^{2} \)
37 \( 1 + 7.24T + 37T^{2} \)
41 \( 1 - 10.3iT - 41T^{2} \)
43 \( 1 - 2.02T + 43T^{2} \)
47 \( 1 + (-3.34 + 3.34i)T - 47iT^{2} \)
53 \( 1 + 7.30iT - 53T^{2} \)
59 \( 1 + (-3.52 + 3.52i)T - 59iT^{2} \)
61 \( 1 + (-1.41 - 1.41i)T + 61iT^{2} \)
67 \( 1 + 0.748T + 67T^{2} \)
71 \( 1 + 0.269T + 71T^{2} \)
73 \( 1 + (-0.811 - 0.811i)T + 73iT^{2} \)
79 \( 1 - 2.80T + 79T^{2} \)
83 \( 1 - 12.8iT - 83T^{2} \)
89 \( 1 - 13.3T + 89T^{2} \)
97 \( 1 + (6.33 + 6.33i)T + 97iT^{2} \)
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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

−11.28883823673561362950832364428, −10.19671303593500952225318561123, −9.367127341635412965234065927387, −8.346029772457150238350611412424, −7.65732537651531094134619524882, −6.66532631821139368849508246676, −5.15060679163287718940057636018, −3.94195825482873923241995948147, −2.31227735195356324441565351051, −1.40702270521890838491067950506, 1.22732345438779414211023638819, 3.68698358282493301399778217837, 4.76228176395743325445806955846, 5.53459496623868219442191995809, 7.04596480541897643339897723796, 7.55272528722941571415228201595, 8.797196097633058716162709772352, 9.486237329914763692975247759106, 10.42245988119571511349413807028, 10.97666456306075976881029729314

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