Properties

Label 2-20e2-25.11-c1-0-9
Degree $2$
Conductor $400$
Sign $0.0948 + 0.995i$
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  + (0.0461 − 0.141i)3-s + (0.383 − 2.20i)5-s − 3.58·7-s + (2.40 + 1.75i)9-s + (4.77 − 3.46i)11-s + (−3.77 − 2.74i)13-s + (−0.295 − 0.156i)15-s + (−1.19 − 3.67i)17-s + (−0.680 − 2.09i)19-s + (−0.165 + 0.509i)21-s + (2.97 − 2.16i)23-s + (−4.70 − 1.69i)25-s + (0.721 − 0.524i)27-s + (−0.982 + 3.02i)29-s + (−1.10 − 3.39i)31-s + ⋯
L(s)  = 1  + (0.0266 − 0.0819i)3-s + (0.171 − 0.985i)5-s − 1.35·7-s + (0.803 + 0.583i)9-s + (1.43 − 1.04i)11-s + (−1.04 − 0.760i)13-s + (−0.0761 − 0.0402i)15-s + (−0.289 − 0.890i)17-s + (−0.156 − 0.480i)19-s + (−0.0361 + 0.111i)21-s + (0.620 − 0.450i)23-s + (−0.941 − 0.338i)25-s + (0.138 − 0.100i)27-s + (−0.182 + 0.561i)29-s + (−0.198 − 0.609i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0948 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0948 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.0948 + 0.995i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.0948 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.900764 - 0.819035i\)
\(L(\frac12)\) \(\approx\) \(0.900764 - 0.819035i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.383 + 2.20i)T \)
good3 \( 1 + (-0.0461 + 0.141i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + 3.58T + 7T^{2} \)
11 \( 1 + (-4.77 + 3.46i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (3.77 + 2.74i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.19 + 3.67i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.680 + 2.09i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-2.97 + 2.16i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (0.982 - 3.02i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.10 + 3.39i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-8.01 - 5.82i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (1.74 + 1.26i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 3.33T + 43T^{2} \)
47 \( 1 + (1.74 - 5.37i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-0.830 + 2.55i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-3.72 - 2.70i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-3.08 + 2.23i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-2.97 - 9.16i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (2.08 - 6.42i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-6.72 + 4.88i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.83 - 8.72i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-0.500 - 1.54i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (5.59 - 4.06i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (3.70 - 11.4i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.11171068855446382021890095085, −9.808996830373429208267950311759, −9.418009939865029344166925956049, −8.446984676784799221502400563241, −7.21091481989112952901342274838, −6.35712318709973740757671598884, −5.19972758492546390383389547841, −4.11831831162973258570600116987, −2.75207569602248002702823967540, −0.826512997043015275928671102059, 1.99080681393859835604428320973, 3.51537242264570302436303846127, 4.28875529383788891073288300020, 6.16320765933426447645592628349, 6.77622742330932805839492425408, 7.34782991255542477549382893844, 9.232508679189966772520539559234, 9.635336295831580563157851435679, 10.31126671265910555731330490790, 11.57124278377304254800609877914

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