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Properties

Label 2-1040-13.4-c1-0-25
Degree $2$
Conductor $1040$
Sign $-0.566 + 0.823i$
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  + (1.41 − 2.44i)3-s + i·5-s + (1.81 − 1.04i)7-s + (−2.49 − 4.32i)9-s + (−1.5 − 0.866i)11-s + (−3.59 − 0.331i)13-s + (2.44 + 1.41i)15-s + (−1.81 − 3.14i)17-s + (−0.926 + 0.534i)19-s − 5.92i·21-s + (3.90 − 6.77i)23-s − 25-s − 5.62·27-s + (0.263 − 0.456i)29-s − 5.84i·31-s + ⋯
L(s)  = 1  + (0.816 − 1.41i)3-s + 0.447i·5-s + (0.685 − 0.395i)7-s + (−0.831 − 1.44i)9-s + (−0.452 − 0.261i)11-s + (−0.995 − 0.0918i)13-s + (0.632 + 0.364i)15-s + (−0.439 − 0.762i)17-s + (−0.212 + 0.122i)19-s − 1.29i·21-s + (0.815 − 1.41i)23-s − 0.200·25-s − 1.08·27-s + (0.0489 − 0.0847i)29-s − 1.04i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.566 + 0.823i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.566 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $-0.566 + 0.823i$
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: \((2,\ 1040,\ (\ :1/2),\ -0.566 + 0.823i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.918525547\)
\(L(\frac12)\) \(\approx\) \(1.918525547\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.59 + 0.331i)T \)
good3 \( 1 + (-1.41 + 2.44i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (-1.81 + 1.04i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.5 + 0.866i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.81 + 3.14i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.926 - 0.534i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.90 + 6.77i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.263 + 0.456i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 5.84iT - 31T^{2} \)
37 \( 1 + (-8.44 - 4.87i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.69 + 2.13i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.67 + 8.09i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 3.46iT - 47T^{2} \)
53 \( 1 - 12.5T + 53T^{2} \)
59 \( 1 + (-1.21 + 0.701i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.55 - 9.62i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.38 - 5.41i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-12.2 + 7.08i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 2.64iT - 73T^{2} \)
79 \( 1 + 13.5T + 79T^{2} \)
83 \( 1 - 15.7iT - 83T^{2} \)
89 \( 1 + (4.78 + 2.76i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (13.1 - 7.59i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.519473040448850830008604068153, −8.447717688971622946446111958354, −7.988665215493599453751770203798, −7.09347458137395811181598009581, −6.72766333890557148871273128383, −5.40548259058762029136271486608, −4.25826822564672324604899385381, −2.75457724066815897781714903452, −2.29474844216893159975652700035, −0.77130283353954285850175047180, 1.96024807637463883822923120689, 3.04427102771055338799100819161, 4.13164007641375914318048053876, 4.91836599375385645294722950280, 5.44931148815307945234468952273, 7.03347797223864638580356953475, 8.131169944813664985315824705173, 8.582220259773522711298325188704, 9.509837086789720123840248207242, 9.916648569004739747599201140994

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