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Properties

Label 2-1040-65.64-c1-0-20
Degree $2$
Conductor $1040$
Sign $0.519 - 0.854i$
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  + 1.57i·3-s + (2.14 − 0.629i)5-s + 4.19·7-s + 0.528·9-s + 6.07i·11-s + (−2.66 + 2.43i)13-s + (0.988 + 3.37i)15-s − 6.59i·17-s − 4.12i·19-s + 6.59i·21-s + 0.313i·23-s + (4.20 − 2.69i)25-s + 5.54i·27-s + 1.50·29-s + 0.732i·31-s + ⋯
L(s)  = 1  + 0.907i·3-s + (0.959 − 0.281i)5-s + 1.58·7-s + 0.176·9-s + 1.83i·11-s + (−0.738 + 0.674i)13-s + (0.255 + 0.870i)15-s − 1.60i·17-s − 0.946i·19-s + 1.43i·21-s + 0.0654i·23-s + (0.841 − 0.539i)25-s + 1.06i·27-s + 0.279·29-s + 0.131i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.519 - 0.854i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.519 - 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $0.519 - 0.854i$
Analytic conductor: \(8.30444\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1040} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1040,\ (\ :1/2),\ 0.519 - 0.854i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.297738929\)
\(L(\frac12)\) \(\approx\) \(2.297738929\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.14 + 0.629i)T \)
13 \( 1 + (2.66 - 2.43i)T \)
good3 \( 1 - 1.57iT - 3T^{2} \)
7 \( 1 - 4.19T + 7T^{2} \)
11 \( 1 - 6.07iT - 11T^{2} \)
17 \( 1 + 6.59iT - 17T^{2} \)
19 \( 1 + 4.12iT - 19T^{2} \)
23 \( 1 - 0.313iT - 23T^{2} \)
29 \( 1 - 1.50T + 29T^{2} \)
31 \( 1 - 0.732iT - 31T^{2} \)
37 \( 1 + 6.19T + 37T^{2} \)
41 \( 1 + 1.19iT - 41T^{2} \)
43 \( 1 + 8.62iT - 43T^{2} \)
47 \( 1 + 4.19T + 47T^{2} \)
53 \( 1 + 3.54iT - 53T^{2} \)
59 \( 1 - 6.55iT - 59T^{2} \)
61 \( 1 + 1.69T + 61T^{2} \)
67 \( 1 + 6.21T + 67T^{2} \)
71 \( 1 - 5.23iT - 71T^{2} \)
73 \( 1 - 4.21T + 73T^{2} \)
79 \( 1 + 11.5T + 79T^{2} \)
83 \( 1 + 8.29T + 83T^{2} \)
89 \( 1 - 13.4iT - 89T^{2} \)
97 \( 1 - 10.2T + 97T^{2} \)
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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

−9.958796379442827599635987774234, −9.378733068076802094108389838000, −8.733560543965293281018226852588, −7.31648202510555068803795348095, −6.99659862809212277374692223971, −5.16557399185619102004868724540, −4.92360920104540506354207789989, −4.32485379023192086678258206494, −2.45739461101794668605610897778, −1.62861498094427358736589352617, 1.24055711334998050469884817195, 1.96726787009000147429073562949, 3.27727635828882598414171702766, 4.70199358581093308356773036042, 5.79174931962936175261933374179, 6.18632535214251952599847286969, 7.41402983030902505426269836333, 8.182412095724162362975224850972, 8.580159504733715818081838921505, 9.985847339171793061459069575056

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