Properties

Label 2-20e2-400.147-c1-0-1
Degree $2$
Conductor $400$
Sign $-0.644 - 0.764i$
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  + (−0.286 − 1.38i)2-s + (−0.553 + 0.761i)3-s + (−1.83 + 0.793i)4-s + (1.47 + 1.68i)5-s + (1.21 + 0.548i)6-s + (−2.97 − 2.97i)7-s + (1.62 + 2.31i)8-s + (0.653 + 2.01i)9-s + (1.90 − 2.52i)10-s + (−4.43 + 2.26i)11-s + (0.411 − 1.83i)12-s + (−1.80 − 5.55i)13-s + (−3.26 + 4.96i)14-s + (−2.09 + 0.193i)15-s + (2.74 − 2.91i)16-s + (−3.45 − 0.547i)17-s + ⋯
L(s)  = 1  + (−0.202 − 0.979i)2-s + (−0.319 + 0.439i)3-s + (−0.918 + 0.396i)4-s + (0.659 + 0.751i)5-s + (0.495 + 0.223i)6-s + (−1.12 − 1.12i)7-s + (0.574 + 0.818i)8-s + (0.217 + 0.670i)9-s + (0.602 − 0.798i)10-s + (−1.33 + 0.681i)11-s + (0.118 − 0.530i)12-s + (−0.500 − 1.53i)13-s + (−0.872 + 1.32i)14-s + (−0.541 + 0.0499i)15-s + (0.685 − 0.728i)16-s + (−0.837 − 0.132i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.644 - 0.764i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.644 - 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.644 - 0.764i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ -0.644 - 0.764i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0632277 + 0.136099i\)
\(L(\frac12)\) \(\approx\) \(0.0632277 + 0.136099i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.286 + 1.38i)T \)
5 \( 1 + (-1.47 - 1.68i)T \)
good3 \( 1 + (0.553 - 0.761i)T + (-0.927 - 2.85i)T^{2} \)
7 \( 1 + (2.97 + 2.97i)T + 7iT^{2} \)
11 \( 1 + (4.43 - 2.26i)T + (6.46 - 8.89i)T^{2} \)
13 \( 1 + (1.80 + 5.55i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (3.45 + 0.547i)T + (16.1 + 5.25i)T^{2} \)
19 \( 1 + (6.21 + 0.984i)T + (18.0 + 5.87i)T^{2} \)
23 \( 1 + (-3.63 - 7.13i)T + (-13.5 + 18.6i)T^{2} \)
29 \( 1 + (2.43 - 0.385i)T + (27.5 - 8.96i)T^{2} \)
31 \( 1 + (0.486 + 0.670i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.910 - 2.80i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-3.03 + 0.985i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + 2.12T + 43T^{2} \)
47 \( 1 + (-2.49 + 0.394i)T + (44.6 - 14.5i)T^{2} \)
53 \( 1 + (1.25 - 1.72i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.04 - 0.531i)T + (34.6 + 47.7i)T^{2} \)
61 \( 1 + (7.03 - 3.58i)T + (35.8 - 49.3i)T^{2} \)
67 \( 1 + (-10.6 + 7.71i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (2.82 + 2.05i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.292 + 0.149i)T + (42.9 - 59.0i)T^{2} \)
79 \( 1 + (-5.70 - 4.14i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (4.22 + 5.81i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (-2.16 + 6.64i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (1.48 + 9.39i)T + (-92.2 + 29.9i)T^{2} \)
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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

−11.00970243279535427809076649376, −10.62456876030531529504097658808, −10.10763574279074982542582949850, −9.457935196605845403532578779891, −7.83531786808038796651942540525, −7.12417059826985273795349484276, −5.59617986378700850054686622228, −4.61469874603499878253565902029, −3.30050380871823512904787130773, −2.31243319986752882157671386715, 0.10070758249843302729906313839, 2.30355064582257836474512453139, 4.33292282046400920417294933500, 5.47887291337856376145742776249, 6.33166694688024740006094610661, 6.75987103732661931867140903252, 8.373404602920825848650747490604, 9.028131601841175644169210278733, 9.595802588294567193102255027079, 10.76464047621901932349521346883

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