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Properties

Label 2-1040-5.4-c1-0-5
Degree $2$
Conductor $1040$
Sign $-0.139 - 0.990i$
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  + 1.31i·3-s + (−2.21 + 0.311i)5-s − 2.90i·7-s + 1.28·9-s − 0.214·11-s + i·13-s + (−0.407 − 2.90i)15-s + 6.42i·17-s + 2.21·19-s + 3.80·21-s + 4.68i·23-s + (4.80 − 1.37i)25-s + 5.61i·27-s − 8.70·29-s + 5.59·31-s + ⋯
L(s)  = 1  + 0.756i·3-s + (−0.990 + 0.139i)5-s − 1.09i·7-s + 0.426·9-s − 0.0646·11-s + 0.277i·13-s + (−0.105 − 0.749i)15-s + 1.55i·17-s + 0.507·19-s + 0.830·21-s + 0.977i·23-s + (0.961 − 0.275i)25-s + 1.08i·27-s − 1.61·29-s + 1.00·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.139 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.139 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $-0.139 - 0.990i$
Analytic conductor: \(8.30444\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1040} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1040,\ (\ :1/2),\ -0.139 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.159061534\)
\(L(\frac12)\) \(\approx\) \(1.159061534\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.21 - 0.311i)T \)
13 \( 1 - iT \)
good3 \( 1 - 1.31iT - 3T^{2} \)
7 \( 1 + 2.90iT - 7T^{2} \)
11 \( 1 + 0.214T + 11T^{2} \)
17 \( 1 - 6.42iT - 17T^{2} \)
19 \( 1 - 2.21T + 19T^{2} \)
23 \( 1 - 4.68iT - 23T^{2} \)
29 \( 1 + 8.70T + 29T^{2} \)
31 \( 1 - 5.59T + 31T^{2} \)
37 \( 1 - 2.28iT - 37T^{2} \)
41 \( 1 - 3.05T + 41T^{2} \)
43 \( 1 - 6.36iT - 43T^{2} \)
47 \( 1 + 1.09iT - 47T^{2} \)
53 \( 1 - 6.23iT - 53T^{2} \)
59 \( 1 + 9.26T + 59T^{2} \)
61 \( 1 + 0.280T + 61T^{2} \)
67 \( 1 - 7.76iT - 67T^{2} \)
71 \( 1 - 6.08T + 71T^{2} \)
73 \( 1 - 10.2iT - 73T^{2} \)
79 \( 1 + 14.2T + 79T^{2} \)
83 \( 1 - 9.52iT - 83T^{2} \)
89 \( 1 - 5.61T + 89T^{2} \)
97 \( 1 + 18.0iT - 97T^{2} \)
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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

−10.17538725537357322610398866303, −9.524499453377073173395275958829, −8.433243047079962266144091958289, −7.58115980097149871070812588840, −7.04656124762028306515700411199, −5.83466347646454114385792631286, −4.54013527919176483414552820632, −4.02804092254574906469736593587, −3.30884605267189058214957924083, −1.34906994787743329668890711868, 0.58082173087084047391384731892, 2.17976795090515919758594184309, 3.21270316226796421291761161133, 4.50020887407398394142107565921, 5.36388015962917131827098926333, 6.45280429550479773810564806759, 7.36267295893826392128188872693, 7.84695286788277299753861944275, 8.830659788186229786920714312963, 9.488580750785328292461459152884

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