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Properties

Label 2-1040-13.10-c1-0-8
Degree $2$
Conductor $1040$
Sign $-0.489 - 0.872i$
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  + (1.05 + 1.82i)3-s + i·5-s + (2.22 + 1.28i)7-s + (−0.717 + 1.24i)9-s + (−0.677 + 0.391i)11-s + (−3.60 + 0.00176i)13-s + (−1.82 + 1.05i)15-s + (−2.71 + 4.70i)17-s + (2.95 + 1.70i)19-s + 5.40i·21-s + (1.35 + 2.33i)23-s − 25-s + 3.29·27-s + (1.43 + 2.48i)29-s − 4.66i·31-s + ⋯
L(s)  = 1  + (0.607 + 1.05i)3-s + 0.447i·5-s + (0.840 + 0.485i)7-s + (−0.239 + 0.414i)9-s + (−0.204 + 0.117i)11-s + (−0.999 + 0.000490i)13-s + (−0.470 + 0.271i)15-s + (−0.658 + 1.14i)17-s + (0.677 + 0.391i)19-s + 1.18i·21-s + (0.281 + 0.487i)23-s − 0.200·25-s + 0.634·27-s + (0.266 + 0.461i)29-s − 0.838i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.489 - 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.489 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $-0.489 - 0.872i$
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: \((2,\ 1040,\ (\ :1/2),\ -0.489 - 0.872i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.941854719\)
\(L(\frac12)\) \(\approx\) \(1.941854719\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.60 - 0.00176i)T \)
good3 \( 1 + (-1.05 - 1.82i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-2.22 - 1.28i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.677 - 0.391i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.71 - 4.70i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.95 - 1.70i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.35 - 2.33i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.43 - 2.48i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.66iT - 31T^{2} \)
37 \( 1 + (4.67 - 2.69i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.54 + 2.04i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.06 + 3.57i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.17iT - 47T^{2} \)
53 \( 1 + 7.43T + 53T^{2} \)
59 \( 1 + (-11.6 - 6.72i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.200 + 0.347i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (11.8 - 6.84i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.07 + 2.35i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 7.33iT - 73T^{2} \)
79 \( 1 - 8.10T + 79T^{2} \)
83 \( 1 - 14.2iT - 83T^{2} \)
89 \( 1 + (-11.5 + 6.65i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.98 - 1.72i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.17571588972260413166456274697, −9.398384069949609215883440941228, −8.660649184609190072271383613396, −7.88115755135265426024653488792, −6.96857058619894779086840944381, −5.72088861936471450061607013271, −4.85825987639616060920795358644, −4.01714757697259216639515425827, −3.00667656023757436189732133848, −1.93586101637184683699111916361, 0.825371299803603391437084138966, 2.04595486388894785260706922479, 2.96887866413347988828853327430, 4.59559370635780452279567935035, 5.08999558896499106058631291770, 6.56319521015045803126609494664, 7.38969174163250297341161614981, 7.79278610374509520338138719122, 8.703671338118574388376051217999, 9.438047054268041331902757463613

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