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Properties

Label 2-1040-13.4-c1-0-20
Degree $2$
Conductor $1040$
Sign $-0.331 + 0.943i$
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  + (−0.800 + 1.38i)3-s + i·5-s + (0.287 − 0.166i)7-s + (0.219 + 0.380i)9-s + (−4.65 − 2.68i)11-s + (−3.55 + 0.619i)13-s + (−1.38 − 0.800i)15-s + (−2.53 − 4.38i)17-s + (1.96 − 1.13i)19-s + 0.531i·21-s + (1.41 − 2.45i)23-s − 25-s − 5.50·27-s + (1.45 − 2.51i)29-s + 5.46i·31-s + ⋯
L(s)  = 1  + (−0.461 + 0.800i)3-s + 0.447i·5-s + (0.108 − 0.0627i)7-s + (0.0732 + 0.126i)9-s + (−1.40 − 0.809i)11-s + (−0.985 + 0.171i)13-s + (−0.357 − 0.206i)15-s + (−0.614 − 1.06i)17-s + (0.450 − 0.260i)19-s + 0.116i·21-s + (0.296 − 0.512i)23-s − 0.200·25-s − 1.05·27-s + (0.269 − 0.466i)29-s + 0.981i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.331 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $-0.331 + 0.943i$
Analytic conductor: \(8.30444\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1040} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1040,\ (\ :1/2),\ -0.331 + 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2240515151\)
\(L(\frac12)\) \(\approx\) \(0.2240515151\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.55 - 0.619i)T \)
good3 \( 1 + (0.800 - 1.38i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (-0.287 + 0.166i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (4.65 + 2.68i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.53 + 4.38i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.96 + 1.13i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.41 + 2.45i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.45 + 2.51i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.46iT - 31T^{2} \)
37 \( 1 + (5.17 + 2.98i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.23 - 1.86i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.53 - 4.38i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 8.34iT - 47T^{2} \)
53 \( 1 + 1.56T + 53T^{2} \)
59 \( 1 + (2.34 - 1.35i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.05 + 12.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.94 + 5.16i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-11.0 + 6.39i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 9.68iT - 73T^{2} \)
79 \( 1 + 4.51T + 79T^{2} \)
83 \( 1 + 4.26iT - 83T^{2} \)
89 \( 1 + (2.79 + 1.61i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.17 + 1.25i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.821671456790907094989928924299, −9.045598489372739635146152855552, −7.87999377134491856383884362706, −7.26113901883055119512876042068, −6.15521647048760399454957239471, −4.99396385666083357336167494871, −4.80026517947522315165835335506, −3.27403650904458127801091501272, −2.37779952494151410834908814721, −0.10290641660634575931098308049, 1.52034302445759226933262187715, 2.60695480354005172382055438813, 4.14721909115691663958618109932, 5.15374380717296790716730647539, 5.84381228642133111094108556384, 6.98658698776599346574738423758, 7.55458398524818632772315204991, 8.326823267225516847544263071090, 9.450620476411828842200462472252, 10.14953774642095029803088052375

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