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Properties

Label 2-20e2-16.13-c1-0-0
Degree $2$
Conductor $400$
Sign $0.633 - 0.773i$
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  + (0.376 − 1.36i)2-s + (−1.82 − 1.82i)3-s + (−1.71 − 1.02i)4-s + (−3.18 + 1.80i)6-s + 4.50i·7-s + (−2.04 + 1.95i)8-s + 3.68i·9-s + (−1.64 + 1.64i)11-s + (1.25 + 5.01i)12-s + (−1.51 − 1.51i)13-s + (6.14 + 1.69i)14-s + (1.88 + 3.52i)16-s − 1.45·17-s + (5.01 + 1.38i)18-s + (−2.67 − 2.67i)19-s + ⋯
L(s)  = 1  + (0.266 − 0.963i)2-s + (−1.05 − 1.05i)3-s + (−0.857 − 0.513i)4-s + (−1.29 + 0.735i)6-s + 1.70i·7-s + (−0.723 + 0.689i)8-s + 1.22i·9-s + (−0.494 + 0.494i)11-s + (0.363 + 1.44i)12-s + (−0.421 − 0.421i)13-s + (1.64 + 0.454i)14-s + (0.472 + 0.881i)16-s − 0.353·17-s + (1.18 + 0.326i)18-s + (−0.614 − 0.614i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.633 - 0.773i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.633 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.633 - 0.773i$
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: \((2,\ 400,\ (\ :1/2),\ 0.633 - 0.773i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.224288 + 0.106192i\)
\(L(\frac12)\) \(\approx\) \(0.224288 + 0.106192i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.376 + 1.36i)T \)
5 \( 1 \)
good3 \( 1 + (1.82 + 1.82i)T + 3iT^{2} \)
7 \( 1 - 4.50iT - 7T^{2} \)
11 \( 1 + (1.64 - 1.64i)T - 11iT^{2} \)
13 \( 1 + (1.51 + 1.51i)T + 13iT^{2} \)
17 \( 1 + 1.45T + 17T^{2} \)
19 \( 1 + (2.67 + 2.67i)T + 19iT^{2} \)
23 \( 1 - 2.37iT - 23T^{2} \)
29 \( 1 + (-0.924 - 0.924i)T + 29iT^{2} \)
31 \( 1 + 7.20T + 31T^{2} \)
37 \( 1 + (-5.21 + 5.21i)T - 37iT^{2} \)
41 \( 1 - 6.41iT - 41T^{2} \)
43 \( 1 + (7.65 - 7.65i)T - 43iT^{2} \)
47 \( 1 - 2.51T + 47T^{2} \)
53 \( 1 + (1.50 - 1.50i)T - 53iT^{2} \)
59 \( 1 + (5.31 - 5.31i)T - 59iT^{2} \)
61 \( 1 + (1.02 + 1.02i)T + 61iT^{2} \)
67 \( 1 + (5.22 + 5.22i)T + 67iT^{2} \)
71 \( 1 - 1.92iT - 71T^{2} \)
73 \( 1 + 1.39iT - 73T^{2} \)
79 \( 1 - 5.06T + 79T^{2} \)
83 \( 1 + (-2.44 - 2.44i)T + 83iT^{2} \)
89 \( 1 + 9.36iT - 89T^{2} \)
97 \( 1 + 18.6T + 97T^{2} \)
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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

−11.48492925275502448049342717220, −10.97696227496907068149038220026, −9.704117280558180069409079597937, −8.831653395913919970795109823935, −7.67946657075397446604720897083, −6.32540575375630701193985951078, −5.56980876565329715445271060245, −4.79968599711063442381455597965, −2.78628769247216247978431154356, −1.80879564688642905274259567118, 0.16501695708597058253017260124, 3.70912024964294075148274077504, 4.39321456683732643750881517954, 5.25264512120426545523857879256, 6.32349267533710628079197548170, 7.17229514400609593212478549348, 8.213028347367974098957258131083, 9.467395808870675386987919794142, 10.35622998364215188053384008557, 10.86232957688240405589924018164

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