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Properties

Label 2-40e2-20.7-c1-0-0
Degree $2$
Conductor $1600$
Sign $-0.899 + 0.437i$
Analytic cond. $12.7760$
Root an. cond. $3.57436$
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  + (0.707 + 0.707i)3-s + (−2.82 + 2.82i)7-s − 1.99i·9-s + 5.19i·11-s + (−2.44 + 2.44i)13-s + (−3.67 − 3.67i)17-s + 1.73·19-s − 4.00·21-s + (−4.24 − 4.24i)23-s + (3.53 − 3.53i)27-s − 6i·29-s − 3.46i·31-s + (−3.67 + 3.67i)33-s − 3.46·39-s − 3·41-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (−1.06 + 1.06i)7-s − 0.666i·9-s + 1.56i·11-s + (−0.679 + 0.679i)13-s + (−0.891 − 0.891i)17-s + 0.397·19-s − 0.872·21-s + (−0.884 − 0.884i)23-s + (0.680 − 0.680i)27-s − 1.11i·29-s − 0.622i·31-s + (−0.639 + 0.639i)33-s − 0.554·39-s − 0.468·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.899 + 0.437i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.899 + 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.899 + 0.437i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.899 + 0.437i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2384585615\)
\(L(\frac12)\) \(\approx\) \(0.2384585615\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3 \( 1 + (-0.707 - 0.707i)T + 3iT^{2} \)
7 \( 1 + (2.82 - 2.82i)T - 7iT^{2} \)
11 \( 1 - 5.19iT - 11T^{2} \)
13 \( 1 + (2.44 - 2.44i)T - 13iT^{2} \)
17 \( 1 + (3.67 + 3.67i)T + 17iT^{2} \)
19 \( 1 - 1.73T + 19T^{2} \)
23 \( 1 + (4.24 + 4.24i)T + 23iT^{2} \)
29 \( 1 + 6iT - 29T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 + 37iT^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 + (2.82 + 2.82i)T + 43iT^{2} \)
47 \( 1 + (-4.24 + 4.24i)T - 47iT^{2} \)
53 \( 1 + (7.34 - 7.34i)T - 53iT^{2} \)
59 \( 1 + 10.3T + 59T^{2} \)
61 \( 1 - 4T + 61T^{2} \)
67 \( 1 + (4.94 - 4.94i)T - 67iT^{2} \)
71 \( 1 - 10.3iT - 71T^{2} \)
73 \( 1 + (11.0 - 11.0i)T - 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (-2.12 - 2.12i)T + 83iT^{2} \)
89 \( 1 + 3iT - 89T^{2} \)
97 \( 1 + (4.89 + 4.89i)T + 97iT^{2} \)
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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

−9.658512572779348690002907884388, −9.304227682241307465134178192161, −8.542705544660088959485285658704, −7.32512489157396519780023361547, −6.67981069131978467015572039665, −5.90730997550882402611996027842, −4.69525858679575286985239336140, −4.07256841897460576515650563279, −2.77480431698165888056412435945, −2.20620660610336534783670515054, 0.081474071634927170735122506685, 1.56731301217924907622284017238, 3.04553495819540601820870495493, 3.50038738971508459496419983257, 4.75854602105141345118225159476, 5.83236421905734433981334098775, 6.59091829940006503899221705769, 7.45485212098929404716440684579, 8.062233410966065524528287023564, 8.858740667711278582659464329355

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