Properties

Label 2-1040-65.29-c1-0-13
Degree $2$
Conductor $1040$
Sign $0.719 - 0.694i$
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 + (4.00 − 4.49i)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 + (1.03 − 1.16i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.719 - 0.694i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.719 - 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $0.719 - 0.694i$
Analytic conductor: \(8.30444\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1040} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1040,\ (\ :1/2),\ 0.719 - 0.694i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8761241229\)
\(L(\frac12)\) \(\approx\) \(0.8761241229\)
\(L(\frac{3}{2})\) 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

−10.49976477978815702746225187231, −9.230490046595679510715611823492, −8.253136194156857659868159161097, −7.68063688335749702359885269787, −6.39451317278763570312586020091, −5.50754430678320184090835449909, −5.01344079607678874233605796653, −4.00748781372357532878850958200, −2.90078242788430391758831434452, −0.792197199837449202727660201934, 0.790275786605588814677851577688, 1.86579009227951594763340139866, 3.87533951912326488376339086050, 4.78315655191089531493262187763, 5.35371077548230943942145488032, 6.61183245186455288386798819367, 7.36527699047692441470938109019, 7.77727391363411594207273158841, 8.832268339458207986855139533584, 10.10392626298655817191125548388

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