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Properties

Label 2-20e2-80.29-c1-0-1
Degree $2$
Conductor $400$
Sign $-0.0857 - 0.996i$
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  + (−0.0554 − 1.41i)2-s + (0.488 − 0.488i)3-s + (−1.99 + 0.156i)4-s + (−0.717 − 0.663i)6-s − 4.71·7-s + (0.331 + 2.80i)8-s + 2.52i·9-s + (−3.91 + 3.91i)11-s + (−0.897 + 1.05i)12-s + (0.0878 − 0.0878i)13-s + (0.261 + 6.66i)14-s + (3.95 − 0.624i)16-s − 4.67i·17-s + (3.56 − 0.139i)18-s + (−1.81 − 1.81i)19-s + ⋯
L(s)  = 1  + (−0.0391 − 0.999i)2-s + (0.282 − 0.282i)3-s + (−0.996 + 0.0783i)4-s + (−0.292 − 0.270i)6-s − 1.78·7-s + (0.117 + 0.993i)8-s + 0.840i·9-s + (−1.17 + 1.17i)11-s + (−0.259 + 0.303i)12-s + (0.0243 − 0.0243i)13-s + (0.0698 + 1.78i)14-s + (0.987 − 0.156i)16-s − 1.13i·17-s + (0.840 − 0.0329i)18-s + (−0.415 − 0.415i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0857 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0857 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.0857 - 0.996i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ -0.0857 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.109186 + 0.118988i\)
\(L(\frac12)\) \(\approx\) \(0.109186 + 0.118988i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0554 + 1.41i)T \)
5 \( 1 \)
good3 \( 1 + (-0.488 + 0.488i)T - 3iT^{2} \)
7 \( 1 + 4.71T + 7T^{2} \)
11 \( 1 + (3.91 - 3.91i)T - 11iT^{2} \)
13 \( 1 + (-0.0878 + 0.0878i)T - 13iT^{2} \)
17 \( 1 + 4.67iT - 17T^{2} \)
19 \( 1 + (1.81 + 1.81i)T + 19iT^{2} \)
23 \( 1 + 1.63T + 23T^{2} \)
29 \( 1 + (3.26 + 3.26i)T + 29iT^{2} \)
31 \( 1 + 2.12T + 31T^{2} \)
37 \( 1 + (3.97 + 3.97i)T + 37iT^{2} \)
41 \( 1 - 8.25iT - 41T^{2} \)
43 \( 1 + (-2.27 - 2.27i)T + 43iT^{2} \)
47 \( 1 - 4.06iT - 47T^{2} \)
53 \( 1 + (5.03 + 5.03i)T + 53iT^{2} \)
59 \( 1 + (-5.16 + 5.16i)T - 59iT^{2} \)
61 \( 1 + (-7.12 - 7.12i)T + 61iT^{2} \)
67 \( 1 + (7.49 - 7.49i)T - 67iT^{2} \)
71 \( 1 + 4.54iT - 71T^{2} \)
73 \( 1 + 8.30T + 73T^{2} \)
79 \( 1 + 11.5T + 79T^{2} \)
83 \( 1 + (-1.16 + 1.16i)T - 83iT^{2} \)
89 \( 1 - 3.24iT - 89T^{2} \)
97 \( 1 - 13.9iT - 97T^{2} \)
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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

−11.51312577206357534008729456671, −10.41973552553656368247770529356, −9.858075659605313898897781809882, −9.112971360925210105047998838454, −7.86274105532442000670513246344, −7.02614328281094141705754256112, −5.53699819387301153795497602948, −4.44763529805804834754727535664, −3.01550900952779915692437938222, −2.27408651874866890692350536822, 0.097059633956630901485596864358, 3.22334914385565603216229185044, 3.87972734569363166845184595265, 5.64511759712086041401807571025, 6.18685790629610471737590493513, 7.15412320676182626330196462508, 8.416911066898396635191412305001, 8.995937289641008032169211459485, 9.997121669484090281658997912252, 10.57609164231063139600737774930

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