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Properties

Label 2-1040-65.57-c1-0-24
Degree $2$
Conductor $1040$
Sign $0.256 + 0.966i$
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  + (−1.61 + 1.61i)3-s − 2.23i·5-s − 2.23i·9-s + (0.381 + 0.381i)11-s + (−2 + 3i)13-s + (3.61 + 3.61i)15-s + (2.23 − 2.23i)17-s + (0.854 + 0.854i)19-s + (−5.61 − 5.61i)23-s − 5.00·25-s + (−1.23 − 1.23i)27-s − 0.763i·29-s + (0.854 − 0.854i)31-s − 1.23·33-s − 3.70i·37-s + ⋯
L(s)  = 1  + (−0.934 + 0.934i)3-s − 0.999i·5-s − 0.745i·9-s + (0.115 + 0.115i)11-s + (−0.554 + 0.832i)13-s + (0.934 + 0.934i)15-s + (0.542 − 0.542i)17-s + (0.195 + 0.195i)19-s + (−1.17 − 1.17i)23-s − 1.00·25-s + (−0.237 − 0.237i)27-s − 0.141i·29-s + (0.153 − 0.153i)31-s − 0.215·33-s − 0.609i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.256 + 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.256 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $0.256 + 0.966i$
Analytic conductor: \(8.30444\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1040} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1040,\ (\ :1/2),\ 0.256 + 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6952677242\)
\(L(\frac12)\) \(\approx\) \(0.6952677242\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.23iT \)
13 \( 1 + (2 - 3i)T \)
good3 \( 1 + (1.61 - 1.61i)T - 3iT^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + (-0.381 - 0.381i)T + 11iT^{2} \)
17 \( 1 + (-2.23 + 2.23i)T - 17iT^{2} \)
19 \( 1 + (-0.854 - 0.854i)T + 19iT^{2} \)
23 \( 1 + (5.61 + 5.61i)T + 23iT^{2} \)
29 \( 1 + 0.763iT - 29T^{2} \)
31 \( 1 + (-0.854 + 0.854i)T - 31iT^{2} \)
37 \( 1 + 3.70iT - 37T^{2} \)
41 \( 1 + (-8.23 + 8.23i)T - 41iT^{2} \)
43 \( 1 + (2.85 + 2.85i)T + 43iT^{2} \)
47 \( 1 + 8.94iT - 47T^{2} \)
53 \( 1 + (7.47 - 7.47i)T - 53iT^{2} \)
59 \( 1 + (-5.61 + 5.61i)T - 59iT^{2} \)
61 \( 1 + 7.70T + 61T^{2} \)
67 \( 1 - 13.7T + 67T^{2} \)
71 \( 1 + (0.381 - 0.381i)T - 71iT^{2} \)
73 \( 1 + 1.70T + 73T^{2} \)
79 \( 1 + 9.70iT - 79T^{2} \)
83 \( 1 + 1.52iT - 83T^{2} \)
89 \( 1 + (-2.23 + 2.23i)T - 89iT^{2} \)
97 \( 1 + 11.4T + 97T^{2} \)
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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

−9.773827257277526829016363172687, −9.182935056382745900262096724747, −8.216931786449967875288574051971, −7.20693089467598548306593991898, −6.02467744065218847899517373123, −5.34732044159408295351057399081, −4.51890705287388017042728016164, −3.94173857934537120076173115117, −2.12687302943640685184716143562, −0.37912696455664695490149265106, 1.29057871433640462567900423628, 2.66595361741789446766334073877, 3.76061918090873927791586965053, 5.24848484526527852131882999058, 6.02879412332417088385128423416, 6.59750857216681346776066657525, 7.61466502512996569079167125950, 7.945504257696295849603951047867, 9.534843631613916677864680617905, 10.18504746582534055272293354763

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