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Properties

Label 2-1040-20.7-c1-0-33
Degree $2$
Conductor $1040$
Sign $-0.417 + 0.908i$
Analytic cond. $8.30444$
Root an. cond. $2.88174$
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  + (−0.343 − 0.343i)3-s + (0.441 − 2.19i)5-s + (3.35 − 3.35i)7-s − 2.76i·9-s + 0.364i·11-s + (0.707 − 0.707i)13-s + (−0.905 + 0.601i)15-s + (−1.37 − 1.37i)17-s − 2.47·19-s − 2.31·21-s + (3.63 + 3.63i)23-s + (−4.61 − 1.93i)25-s + (−1.98 + 1.98i)27-s + 0.280i·29-s + 9.30i·31-s + ⋯
L(s)  = 1  + (−0.198 − 0.198i)3-s + (0.197 − 0.980i)5-s + (1.26 − 1.26i)7-s − 0.921i·9-s + 0.110i·11-s + (0.196 − 0.196i)13-s + (−0.233 + 0.155i)15-s + (−0.333 − 0.333i)17-s − 0.568·19-s − 0.504·21-s + (0.757 + 0.757i)23-s + (−0.922 − 0.387i)25-s + (−0.381 + 0.381i)27-s + 0.0521i·29-s + 1.67i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.417 + 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.417 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $-0.417 + 0.908i$
Analytic conductor: \(8.30444\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1040} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1040,\ (\ :1/2),\ -0.417 + 0.908i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.661203897\)
\(L(\frac12)\) \(\approx\) \(1.661203897\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.441 + 2.19i)T \)
13 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (0.343 + 0.343i)T + 3iT^{2} \)
7 \( 1 + (-3.35 + 3.35i)T - 7iT^{2} \)
11 \( 1 - 0.364iT - 11T^{2} \)
17 \( 1 + (1.37 + 1.37i)T + 17iT^{2} \)
19 \( 1 + 2.47T + 19T^{2} \)
23 \( 1 + (-3.63 - 3.63i)T + 23iT^{2} \)
29 \( 1 - 0.280iT - 29T^{2} \)
31 \( 1 - 9.30iT - 31T^{2} \)
37 \( 1 + (5.28 + 5.28i)T + 37iT^{2} \)
41 \( 1 - 2.51T + 41T^{2} \)
43 \( 1 + (-7.71 - 7.71i)T + 43iT^{2} \)
47 \( 1 + (-1.27 + 1.27i)T - 47iT^{2} \)
53 \( 1 + (8.30 - 8.30i)T - 53iT^{2} \)
59 \( 1 + 10.1T + 59T^{2} \)
61 \( 1 - 7.78T + 61T^{2} \)
67 \( 1 + (-6.01 + 6.01i)T - 67iT^{2} \)
71 \( 1 + 15.2iT - 71T^{2} \)
73 \( 1 + (-5.72 + 5.72i)T - 73iT^{2} \)
79 \( 1 - 10.9T + 79T^{2} \)
83 \( 1 + (-6.23 - 6.23i)T + 83iT^{2} \)
89 \( 1 + 9.52iT - 89T^{2} \)
97 \( 1 + (-6.96 - 6.96i)T + 97iT^{2} \)
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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

−9.490683590117074660201957761059, −8.893486876153188560777048882024, −7.948012549934928144180328162176, −7.25481808794993235124425731037, −6.28651730998586821540562771780, −5.13865030127231600674207701506, −4.52225655311992161327481368961, −3.53616087038197694082902912503, −1.66840409115693333598181850694, −0.809476725729739153532200877941, 1.98653210153106369461812764373, 2.56688081384873189752726084579, 4.12513618548533887252370149252, 5.11142189461875302537946737431, 5.82906473399071793920842388678, 6.74441041898730292140106106326, 7.86761804863646621858716012271, 8.425779155827813769123005532385, 9.338371170491271295958687227736, 10.40237904202024837805400082716

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