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Properties

Label 2-20e2-16.13-c1-0-11
Degree $2$
Conductor $400$
Sign $0.601 - 0.799i$
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  + (−1.22 + 0.710i)2-s + (1.09 + 1.09i)3-s + (0.991 − 1.73i)4-s + (−2.11 − 0.560i)6-s − 0.973i·7-s + (0.0202 + 2.82i)8-s − 0.616i·9-s + (1.40 − 1.40i)11-s + (2.97 − 0.813i)12-s + (4.60 + 4.60i)13-s + (0.691 + 1.19i)14-s + (−2.03 − 3.44i)16-s − 0.490·17-s + (0.438 + 0.754i)18-s + (4.54 + 4.54i)19-s + ⋯
L(s)  = 1  + (−0.864 + 0.502i)2-s + (0.630 + 0.630i)3-s + (0.495 − 0.868i)4-s + (−0.861 − 0.228i)6-s − 0.368i·7-s + (0.00714 + 0.999i)8-s − 0.205i·9-s + (0.424 − 0.424i)11-s + (0.859 − 0.234i)12-s + (1.27 + 1.27i)13-s + (0.184 + 0.318i)14-s + (−0.508 − 0.861i)16-s − 0.118·17-s + (0.103 + 0.177i)18-s + (1.04 + 1.04i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.601 - 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.601 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.601 - 0.799i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.601 - 0.799i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08997 + 0.543985i\)
\(L(\frac12)\) \(\approx\) \(1.08997 + 0.543985i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 - 0.710i)T \)
5 \( 1 \)
good3 \( 1 + (-1.09 - 1.09i)T + 3iT^{2} \)
7 \( 1 + 0.973iT - 7T^{2} \)
11 \( 1 + (-1.40 + 1.40i)T - 11iT^{2} \)
13 \( 1 + (-4.60 - 4.60i)T + 13iT^{2} \)
17 \( 1 + 0.490T + 17T^{2} \)
19 \( 1 + (-4.54 - 4.54i)T + 19iT^{2} \)
23 \( 1 + 1.94iT - 23T^{2} \)
29 \( 1 + (3.74 + 3.74i)T + 29iT^{2} \)
31 \( 1 - 4.29T + 31T^{2} \)
37 \( 1 + (4.55 - 4.55i)T - 37iT^{2} \)
41 \( 1 - 10.1iT - 41T^{2} \)
43 \( 1 + (-1.79 + 1.79i)T - 43iT^{2} \)
47 \( 1 + 10.0T + 47T^{2} \)
53 \( 1 + (5.61 - 5.61i)T - 53iT^{2} \)
59 \( 1 + (-8.44 + 8.44i)T - 59iT^{2} \)
61 \( 1 + (-3.01 - 3.01i)T + 61iT^{2} \)
67 \( 1 + (7.07 + 7.07i)T + 67iT^{2} \)
71 \( 1 + 0.897iT - 71T^{2} \)
73 \( 1 + 9.71iT - 73T^{2} \)
79 \( 1 + 14.7T + 79T^{2} \)
83 \( 1 + (0.815 + 0.815i)T + 83iT^{2} \)
89 \( 1 + 1.12iT - 89T^{2} \)
97 \( 1 - 7.54T + 97T^{2} \)
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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

−11.24641254275352620447822294882, −10.13769652516747445713390478369, −9.488448873263459027945930130606, −8.717696103169368520598957952422, −8.001745469447698717315220125669, −6.71029107409713533279594682856, −6.01085989980836550008462959181, −4.41838691567650338253669331563, −3.31520959799533586941644600809, −1.42203458514576947960960602343, 1.27860447325928339487701772158, 2.59416463587984407894107794533, 3.61031154101618971058326347838, 5.42930194857329474255322584535, 6.87134552323787431983406374147, 7.60879075738881650166411082057, 8.524536539665312303257429822034, 9.090127556883707601000282370734, 10.22868880820803159403264703619, 11.07377978005446662979867645519

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