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Properties

Label 2-40e2-80.3-c1-0-25
Degree $2$
Conductor $1600$
Sign $-0.227 + 0.973i$
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  + 0.619·3-s + (−1.82 + 1.82i)7-s − 2.61·9-s + (0.567 + 0.567i)11-s − 2.78i·13-s + (3.65 − 3.65i)17-s + (−4.51 − 4.51i)19-s + (−1.12 + 1.12i)21-s + (2.15 + 2.15i)23-s − 3.47·27-s + (−3.20 + 3.20i)29-s − 3.54i·31-s + (0.351 + 0.351i)33-s − 5.22i·37-s − 1.72i·39-s + ⋯
L(s)  = 1  + 0.357·3-s + (−0.689 + 0.689i)7-s − 0.872·9-s + (0.171 + 0.171i)11-s − 0.773i·13-s + (0.885 − 0.885i)17-s + (−1.03 − 1.03i)19-s + (−0.246 + 0.246i)21-s + (0.449 + 0.449i)23-s − 0.669·27-s + (−0.594 + 0.594i)29-s − 0.635i·31-s + (0.0611 + 0.0611i)33-s − 0.858i·37-s − 0.276i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.227 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.227 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.227 + 0.973i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (943, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.227 + 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9683588639\)
\(L(\frac12)\) \(\approx\) \(0.9683588639\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.619T + 3T^{2} \)
7 \( 1 + (1.82 - 1.82i)T - 7iT^{2} \)
11 \( 1 + (-0.567 - 0.567i)T + 11iT^{2} \)
13 \( 1 + 2.78iT - 13T^{2} \)
17 \( 1 + (-3.65 + 3.65i)T - 17iT^{2} \)
19 \( 1 + (4.51 + 4.51i)T + 19iT^{2} \)
23 \( 1 + (-2.15 - 2.15i)T + 23iT^{2} \)
29 \( 1 + (3.20 - 3.20i)T - 29iT^{2} \)
31 \( 1 + 3.54iT - 31T^{2} \)
37 \( 1 + 5.22iT - 37T^{2} \)
41 \( 1 + 8.76iT - 41T^{2} \)
43 \( 1 + 10.8iT - 43T^{2} \)
47 \( 1 + (3.22 + 3.22i)T + 47iT^{2} \)
53 \( 1 - 12.8T + 53T^{2} \)
59 \( 1 + (-3.79 + 3.79i)T - 59iT^{2} \)
61 \( 1 + (-6.63 - 6.63i)T + 61iT^{2} \)
67 \( 1 + 7.78iT - 67T^{2} \)
71 \( 1 + 13.6T + 71T^{2} \)
73 \( 1 + (-1.34 + 1.34i)T - 73iT^{2} \)
79 \( 1 + 16.3T + 79T^{2} \)
83 \( 1 - 0.391T + 83T^{2} \)
89 \( 1 + 18.0T + 89T^{2} \)
97 \( 1 + (6.43 - 6.43i)T - 97iT^{2} \)
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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

−8.932286157692138337341809230805, −8.734584396377183826060567318400, −7.53269931647729817229302272361, −6.88420274263068544399363621813, −5.65626550786269313228545572430, −5.38766954883880253146930718535, −3.90245487186438419728320666529, −2.99152919626005215347648009338, −2.28288686522589683210604174638, −0.35219562763263228436656627995, 1.42541113000447839632027152770, 2.78870283887630978856677068140, 3.66235273851730709869135283546, 4.42205140346305139152550564486, 5.78873878350363265063157602126, 6.34164739704606417644632492691, 7.23333467071192365958281834114, 8.276633651373274323587775003565, 8.629235420831112252906243687602, 9.800337090010040709055148587535

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