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Properties

Label 2-40e2-80.29-c1-0-8
Degree $2$
Conductor $1600$
Sign $-0.388 - 0.921i$
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  + (−0.720 + 0.720i)3-s + 4.02·7-s + 1.96i·9-s + (0.646 − 0.646i)11-s + (−4.91 + 4.91i)13-s + 2.70i·17-s + (−0.438 − 0.438i)19-s + (−2.90 + 2.90i)21-s + 3.60·23-s + (−3.57 − 3.57i)27-s + (−2.00 − 2.00i)29-s − 4.30·31-s + 0.932i·33-s + (0.743 + 0.743i)37-s − 7.08i·39-s + ⋯
L(s)  = 1  + (−0.416 + 0.416i)3-s + 1.52·7-s + 0.653i·9-s + (0.195 − 0.195i)11-s + (−1.36 + 1.36i)13-s + 0.656i·17-s + (−0.100 − 0.100i)19-s + (−0.633 + 0.633i)21-s + 0.750·23-s + (−0.688 − 0.688i)27-s + (−0.373 − 0.373i)29-s − 0.774·31-s + 0.162i·33-s + (0.122 + 0.122i)37-s − 1.13i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.388 - 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.388 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.388 - 0.921i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.388 - 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.365925104\)
\(L(\frac12)\) \(\approx\) \(1.365925104\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.720 - 0.720i)T - 3iT^{2} \)
7 \( 1 - 4.02T + 7T^{2} \)
11 \( 1 + (-0.646 + 0.646i)T - 11iT^{2} \)
13 \( 1 + (4.91 - 4.91i)T - 13iT^{2} \)
17 \( 1 - 2.70iT - 17T^{2} \)
19 \( 1 + (0.438 + 0.438i)T + 19iT^{2} \)
23 \( 1 - 3.60T + 23T^{2} \)
29 \( 1 + (2.00 + 2.00i)T + 29iT^{2} \)
31 \( 1 + 4.30T + 31T^{2} \)
37 \( 1 + (-0.743 - 0.743i)T + 37iT^{2} \)
41 \( 1 + 0.603iT - 41T^{2} \)
43 \( 1 + (-5.03 - 5.03i)T + 43iT^{2} \)
47 \( 1 - 10.8iT - 47T^{2} \)
53 \( 1 + (-4.07 - 4.07i)T + 53iT^{2} \)
59 \( 1 + (-1.22 + 1.22i)T - 59iT^{2} \)
61 \( 1 + (6.98 + 6.98i)T + 61iT^{2} \)
67 \( 1 + (5.24 - 5.24i)T - 67iT^{2} \)
71 \( 1 + 13.7iT - 71T^{2} \)
73 \( 1 - 1.30T + 73T^{2} \)
79 \( 1 - 0.611T + 79T^{2} \)
83 \( 1 + (-1.29 + 1.29i)T - 83iT^{2} \)
89 \( 1 - 10.9iT - 89T^{2} \)
97 \( 1 - 12.7iT - 97T^{2} \)
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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

−9.568918570942707602204310926905, −8.985186729390401042464518895601, −7.895864222830785574130199415691, −7.48715034670512315922463076955, −6.36637424653607665958891127087, −5.31254472119844017544228790929, −4.69877630774113836507035528961, −4.12878627310322997613896275666, −2.44402230617482149446809609832, −1.57072912303703601185959927116, 0.56217919923992016736836681567, 1.79250783913226421021450931265, 2.96768067814035821069045525472, 4.25004672047869875532254905790, 5.25696097529044943617531901947, 5.59831361365197006357315373520, 7.10164954178311106922398836389, 7.32383050064170793378994042515, 8.292828391360531595812149814883, 9.099837934456953161276462530035

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