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Properties

Label 2-40e2-20.7-c1-0-9
Degree $2$
Conductor $1600$
Sign $-0.850 - 0.525i$
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  + (2.23 + 2.23i)3-s + (−2.23 + 2.23i)7-s + 7.00i·9-s − 10.0·21-s + (6.70 + 6.70i)23-s + (−8.94 + 8.94i)27-s − 6i·29-s − 12·41-s + (−2.23 − 2.23i)43-s + (−6.70 + 6.70i)47-s − 3.00i·49-s − 8·61-s + (−15.6 − 15.6i)63-s + (11.1 − 11.1i)67-s + 30.0i·69-s + ⋯
L(s)  = 1  + (1.29 + 1.29i)3-s + (−0.845 + 0.845i)7-s + 2.33i·9-s − 2.18·21-s + (1.39 + 1.39i)23-s + (−1.72 + 1.72i)27-s − 1.11i·29-s − 1.87·41-s + (−0.340 − 0.340i)43-s + (−0.978 + 0.978i)47-s − 0.428i·49-s − 1.02·61-s + (−1.97 − 1.97i)63-s + (1.36 − 1.36i)67-s + 3.61i·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.850 - 0.525i$
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: \((2,\ 1600,\ (\ :1/2),\ -0.850 - 0.525i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.138833643\)
\(L(\frac12)\) \(\approx\) \(2.138833643\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.23 - 2.23i)T + 3iT^{2} \)
7 \( 1 + (2.23 - 2.23i)T - 7iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 13iT^{2} \)
17 \( 1 + 17iT^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + (-6.70 - 6.70i)T + 23iT^{2} \)
29 \( 1 + 6iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 37iT^{2} \)
41 \( 1 + 12T + 41T^{2} \)
43 \( 1 + (2.23 + 2.23i)T + 43iT^{2} \)
47 \( 1 + (6.70 - 6.70i)T - 47iT^{2} \)
53 \( 1 - 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 + (-11.1 + 11.1i)T - 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (-6.70 - 6.70i)T + 83iT^{2} \)
89 \( 1 - 6iT - 89T^{2} \)
97 \( 1 + 97iT^{2} \)
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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

−9.496105630132233981715491433904, −9.185788614370790979489714186462, −8.379010613957239173646645083055, −7.63304384116467567464772599213, −6.49658393546939259146040478205, −5.39534190248615249142837333260, −4.68558965207510125871950860003, −3.49094016899079719483234096769, −3.11583879702628824115326895184, −2.04530559482849895487284918821, 0.68065253205299105847164602824, 1.85059286920267894722971885122, 3.04305013922393692705711668468, 3.49926226400355369727595460109, 4.81285716207612743894987948430, 6.29935147146270531966716659586, 6.94075640828951657934291303632, 7.28034061006716998686749649107, 8.434827604038177592443756923405, 8.745317933887126052803531283058

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