Properties

Label 2-1040-65.9-c1-0-5
Degree $2$
Conductor $1040$
Sign $-0.991 - 0.127i$
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  + (2.33 + 1.34i)3-s + (−2.12 + 0.702i)5-s + (−2.90 + 1.67i)7-s + (2.12 + 3.67i)9-s + (−1.62 + 2.81i)11-s + (−1.21 − 3.39i)13-s + (−5.89 − 1.21i)15-s + (−1.68 + 0.974i)17-s + (0.622 + 1.07i)19-s − 9.02·21-s + (−2.33 − 1.34i)23-s + (4.01 − 2.98i)25-s + 3.35i·27-s + (1.5 − 2.59i)29-s − 3.78·31-s + ⋯
L(s)  = 1  + (1.34 + 0.777i)3-s + (−0.949 + 0.314i)5-s + (−1.09 + 0.633i)7-s + (0.707 + 1.22i)9-s + (−0.489 + 0.847i)11-s + (−0.337 − 0.941i)13-s + (−1.52 − 0.314i)15-s + (−0.409 + 0.236i)17-s + (0.142 + 0.247i)19-s − 1.96·21-s + (−0.486 − 0.280i)23-s + (0.802 − 0.596i)25-s + 0.645i·27-s + (0.278 − 0.482i)29-s − 0.679·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.127i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $-0.991 - 0.127i$
Analytic conductor: \(8.30444\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1040} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1040,\ (\ :1/2),\ -0.991 - 0.127i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.007208820\)
\(L(\frac12)\) \(\approx\) \(1.007208820\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.12 - 0.702i)T \)
13 \( 1 + (1.21 + 3.39i)T \)
good3 \( 1 + (-2.33 - 1.34i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (2.90 - 1.67i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.62 - 2.81i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.68 - 0.974i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.622 - 1.07i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.33 + 1.34i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.78T + 31T^{2} \)
37 \( 1 + (1.68 + 0.974i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.39 - 2.40i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (7.56 - 4.36i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 6.86iT - 47T^{2} \)
53 \( 1 - 12.8iT - 53T^{2} \)
59 \( 1 + (-1.26 - 2.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.74 - 6.48i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.47 + 2.00i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.62 - 4.54i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 5.46iT - 73T^{2} \)
79 \( 1 - 13.7T + 79T^{2} \)
83 \( 1 - 8.61iT - 83T^{2} \)
89 \( 1 + (-5.15 + 8.93i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.56 - 2.63i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.07200844587448307056090514089, −9.555525286539084816290825187424, −8.657118676375645886663625743708, −7.979737959263070909891941719521, −7.27602563889290896839734037459, −6.12661054421170271486569575839, −4.82382332030398054413845785659, −3.92176101526660455933758064540, −3.07140790453103644085971398829, −2.46044794595188180361529456942, 0.35760653168332151520447911181, 2.02359898693882865543271878335, 3.33358027347703448892564524531, 3.67440679048724241384198898490, 5.03964608129877481358664844237, 6.64445239533494609212675733997, 7.06944307686480791996003243885, 7.918325485952659757570139083145, 8.632097730704911204611888292804, 9.239991069137411062457778352505

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