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Properties

Label 2-1040-65.18-c1-0-14
Degree $2$
Conductor $1040$
Sign $0.990 + 0.134i$
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 − i)3-s + (2 + i)5-s + 2·7-s − i·9-s + (1 + i)11-s + (2 + 3i)13-s + (−1 − 3i)15-s + (−1 − i)17-s + (3 + 3i)19-s + (−2 − 2i)21-s + (−1 + i)23-s + (3 + 4i)25-s + (−4 + 4i)27-s − 8i·29-s + (−1 + i)31-s + ⋯
L(s)  = 1  + (−0.577 − 0.577i)3-s + (0.894 + 0.447i)5-s + 0.755·7-s − 0.333i·9-s + (0.301 + 0.301i)11-s + (0.554 + 0.832i)13-s + (−0.258 − 0.774i)15-s + (−0.242 − 0.242i)17-s + (0.688 + 0.688i)19-s + (−0.436 − 0.436i)21-s + (−0.208 + 0.208i)23-s + (0.600 + 0.800i)25-s + (−0.769 + 0.769i)27-s − 1.48i·29-s + (−0.179 + 0.179i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.134i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $0.990 + 0.134i$
Analytic conductor: \(8.30444\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1040} (993, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1040,\ (\ :1/2),\ 0.990 + 0.134i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.771841183\)
\(L(\frac12)\) \(\approx\) \(1.771841183\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 - i)T \)
13 \( 1 + (-2 - 3i)T \)
good3 \( 1 + (1 + i)T + 3iT^{2} \)
7 \( 1 - 2T + 7T^{2} \)
11 \( 1 + (-1 - i)T + 11iT^{2} \)
17 \( 1 + (1 + i)T + 17iT^{2} \)
19 \( 1 + (-3 - 3i)T + 19iT^{2} \)
23 \( 1 + (1 - i)T - 23iT^{2} \)
29 \( 1 + 8iT - 29T^{2} \)
31 \( 1 + (1 - i)T - 31iT^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 + (7 - 7i)T - 41iT^{2} \)
43 \( 1 + (1 - i)T - 43iT^{2} \)
47 \( 1 - 10T + 47T^{2} \)
53 \( 1 + (-1 - i)T + 53iT^{2} \)
59 \( 1 + (-9 + 9i)T - 59iT^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 12iT - 67T^{2} \)
71 \( 1 + (5 - 5i)T - 71iT^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 + 10iT - 79T^{2} \)
83 \( 1 - 18T + 83T^{2} \)
89 \( 1 + (-11 + 11i)T - 89iT^{2} \)
97 \( 1 - 14iT - 97T^{2} \)
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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

−9.783675720916880866307779596830, −9.299927267119627538801175866966, −8.143558487344459696991614498479, −7.23141041349045793084767261005, −6.39595384945979422203430927905, −5.89527287074411596659745698973, −4.82348358580404945025425857462, −3.64844216811107268698501303834, −2.15518095014858741026724343543, −1.24087826438721230777347148708, 1.09890903022930003044714089875, 2.42497372166466164169491189817, 3.89271663717379464718842348369, 5.02701262208173165225212366479, 5.40119533790229489367842510617, 6.29870182968291452155306811500, 7.48422969583501624961889910482, 8.493915870847531530810395120950, 9.087137622755323864098457333040, 10.14270587080169082443167746094

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