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Properties

Label 2-1040-65.9-c1-0-32
Degree $2$
Conductor $1040$
Sign $-0.503 + 0.864i$
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  + (−0.298 − 0.172i)3-s + (1.44 − 1.71i)5-s + (−1.75 + 1.01i)7-s + (−1.44 − 2.49i)9-s + (1.94 − 3.36i)11-s + (2.96 + 2.05i)13-s + (−0.725 + 0.262i)15-s + (−4.71 + 2.72i)17-s + (−2.94 − 5.09i)19-s + 0.700·21-s + (0.298 + 0.172i)23-s + (−0.850 − 4.92i)25-s + 2.02i·27-s + (1.5 − 2.59i)29-s − 1.18·31-s + ⋯
L(s)  = 1  + (−0.172 − 0.0996i)3-s + (0.644 − 0.764i)5-s + (−0.664 + 0.383i)7-s + (−0.480 − 0.831i)9-s + (0.585 − 1.01i)11-s + (0.821 + 0.570i)13-s + (−0.187 + 0.0678i)15-s + (−1.14 + 0.660i)17-s + (−0.674 − 1.16i)19-s + 0.152·21-s + (0.0623 + 0.0359i)23-s + (−0.170 − 0.985i)25-s + 0.390i·27-s + (0.278 − 0.482i)29-s − 0.212·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.503 + 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.503 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $-0.503 + 0.864i$
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: \((2,\ 1040,\ (\ :1/2),\ -0.503 + 0.864i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.144842811\)
\(L(\frac12)\) \(\approx\) \(1.144842811\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.44 + 1.71i)T \)
13 \( 1 + (-2.96 - 2.05i)T \)
good3 \( 1 + (0.298 + 0.172i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (1.75 - 1.01i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.94 + 3.36i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (4.71 - 2.72i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.94 + 5.09i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.298 - 0.172i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.18T + 31T^{2} \)
37 \( 1 + (4.71 + 2.72i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.0902 - 0.156i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.15 - 0.669i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 12.2iT - 47T^{2} \)
53 \( 1 + 2.42iT - 53T^{2} \)
59 \( 1 + (-3.53 - 6.11i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.38 + 5.85i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.81 + 2.20i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.940 + 1.62i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 8.86iT - 73T^{2} \)
79 \( 1 - 11.1T + 79T^{2} \)
83 \( 1 - 7.83iT - 83T^{2} \)
89 \( 1 + (-6.12 + 10.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.02 - 2.90i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.251538393593171800216376390207, −8.955817425051042363261088557495, −8.430236692692469715411712910801, −6.60278790135604881634132899806, −6.38776226293503660473724812184, −5.54647591053456741538431634560, −4.31931195481641245490729975881, −3.34030067339823236069910267667, −1.99107244110755776837276699043, −0.50867349451479540649118703690, 1.77303967059882865097425424468, 2.86767596741634844516185830015, 3.96257931117751282025860915979, 5.07208347660390058591220423646, 6.13166876050631870330971985434, 6.68945558147305500898674090120, 7.57504375263217321354923868919, 8.632932286485085992384017142438, 9.525935907271224928925805724353, 10.33129688705208831364487031111

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