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Properties

Label 2-1040-65.9-c1-0-39
Degree $2$
Conductor $1040$
Sign $-0.984 - 0.175i$
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  + (0.466 + 0.269i)3-s + (−0.539 − 2.17i)5-s + (−0.614 + 0.354i)7-s + (−1.35 − 2.34i)9-s + (−2.25 + 3.90i)11-s + (−3.35 + 1.30i)13-s + (0.333 − 1.15i)15-s + (2.74 − 1.58i)17-s + (0.0603 + 0.104i)19-s − 0.382·21-s + (−4.30 − 2.48i)23-s + (−4.41 + 2.34i)25-s − 3.07i·27-s + (−3.63 + 6.28i)29-s − 9.66·31-s + ⋯
L(s)  = 1  + (0.269 + 0.155i)3-s + (−0.241 − 0.970i)5-s + (−0.232 + 0.134i)7-s + (−0.451 − 0.782i)9-s + (−0.679 + 1.17i)11-s + (−0.931 + 0.362i)13-s + (0.0860 − 0.299i)15-s + (0.665 − 0.384i)17-s + (0.0138 + 0.0239i)19-s − 0.0834·21-s + (−0.897 − 0.518i)23-s + (−0.883 + 0.468i)25-s − 0.592i·27-s + (−0.674 + 1.16i)29-s − 1.73·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.175i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 - 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $-0.984 - 0.175i$
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: \((2,\ 1040,\ (\ :1/2),\ -0.984 - 0.175i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.06952409503\)
\(L(\frac12)\) \(\approx\) \(0.06952409503\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.539 + 2.17i)T \)
13 \( 1 + (3.35 - 1.30i)T \)
good3 \( 1 + (-0.466 - 0.269i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (0.614 - 0.354i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.25 - 3.90i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.74 + 1.58i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.0603 - 0.104i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.30 + 2.48i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.63 - 6.28i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 9.66T + 31T^{2} \)
37 \( 1 + (-6.02 - 3.47i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.223 - 0.387i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.73 + i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 4.70iT - 47T^{2} \)
53 \( 1 + 9.58iT - 53T^{2} \)
59 \( 1 + (-2.87 - 4.98i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.53 + 6.11i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.53 + 1.46i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.09 + 7.08i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 6.74iT - 73T^{2} \)
79 \( 1 + 16.0T + 79T^{2} \)
83 \( 1 + 0.355iT - 83T^{2} \)
89 \( 1 + (-1.81 + 3.14i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.84 - 3.95i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.518885363951399639324136643529, −8.809094132754298725992565132071, −7.80505657077742961181284835255, −7.18505242611990112293670941648, −5.92500185681383866321606757090, −5.04060173537936711270223939350, −4.25196877753748442302538533184, −3.10728947048166516189017070207, −1.84770358027704324316804032008, −0.02770990754544043617470862535, 2.21566571580159108300216176830, 3.04152353778012959434086648151, 3.95292964493205690637204233045, 5.50645821779348248463728792316, 5.93145902610160508270020021363, 7.37310180675698261463549190614, 7.69167762898951520955446091007, 8.511406037860978960232258772691, 9.708050958471778015759113142961, 10.38561909816101775699343230531

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