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Properties

Label 2-1040-13.10-c1-0-20
Degree $2$
Conductor $1040$
Sign $0.111 + 0.993i$
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  + (−0.0473 − 0.0820i)3-s − i·5-s + (−0.716 − 0.413i)7-s + (1.49 − 2.59i)9-s + (−1.5 + 0.866i)11-s + (3.32 + 1.40i)13-s + (−0.0820 + 0.0473i)15-s + (0.716 − 1.24i)17-s + (0.926 + 0.534i)19-s + 0.0783i·21-s + (−1.54 − 2.67i)23-s − 25-s − 0.567·27-s + (−3.72 − 6.45i)29-s − 5.84i·31-s + ⋯
L(s)  = 1  + (−0.0273 − 0.0473i)3-s − 0.447i·5-s + (−0.270 − 0.156i)7-s + (0.498 − 0.863i)9-s + (−0.452 + 0.261i)11-s + (0.921 + 0.388i)13-s + (−0.0211 + 0.0122i)15-s + (0.173 − 0.300i)17-s + (0.212 + 0.122i)19-s + 0.0171i·21-s + (−0.321 − 0.557i)23-s − 0.200·25-s − 0.109·27-s + (−0.692 − 1.19i)29-s − 1.04i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.111 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.111 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $0.111 + 0.993i$
Analytic conductor: \(8.30444\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1040} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1040,\ (\ :1/2),\ 0.111 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.415033795\)
\(L(\frac12)\) \(\approx\) \(1.415033795\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
13 \( 1 + (-3.32 - 1.40i)T \)
good3 \( 1 + (0.0473 + 0.0820i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (0.716 + 0.413i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.5 - 0.866i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.716 + 1.24i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.926 - 0.534i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.54 + 2.67i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.72 + 6.45i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.84iT - 31T^{2} \)
37 \( 1 + (-0.851 + 0.491i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.69 - 2.13i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.77 + 8.26i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.46iT - 47T^{2} \)
53 \( 1 - 0.334T + 53T^{2} \)
59 \( 1 + (-9.98 - 5.76i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.35 - 2.35i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-11.9 + 6.87i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (8.46 + 4.88i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 11.1iT - 73T^{2} \)
79 \( 1 - 0.252T + 79T^{2} \)
83 \( 1 + 5.67iT - 83T^{2} \)
89 \( 1 + (-3.98 + 2.29i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.25 - 4.76i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.720036128388459639853080475536, −8.988376166839944879166031480911, −8.104194977269851341220809516211, −7.19625280047596365661964200774, −6.33171808110879589578901775707, −5.51561156469158322943450285548, −4.28496823927886821986522387705, −3.62046165543929882066615713242, −2.12324156217286009887168946934, −0.67513354212886124539374357974, 1.52085255637884756704482220966, 2.88189292289903954299855121967, 3.77435072064529853104845422634, 5.03260766775419092662729353182, 5.80612155233610674266771428630, 6.78353047566075066630022719682, 7.66903602808589995576514645126, 8.357995166969565749392174481574, 9.354664757292277537864882147441, 10.26485762569828676194764831482

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