↑

Properties

Label 2-1040-260.239-c1-0-5
Degree $2$
Conductor $1040$
Sign $-0.912 - 0.408i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.55·3-s + (−0.824 + 2.07i)5-s + (−1.18 − 1.18i)7-s − 0.592·9-s + (−3.12 + 3.12i)11-s + (−1.19 + 3.40i)13-s + (−1.27 + 3.22i)15-s − 0.389·17-s + (−3.79 − 3.79i)19-s + (−1.83 − 1.83i)21-s − 4.98i·23-s + (−3.63 − 3.42i)25-s − 5.57·27-s − 3.73·29-s + (5.94 + 5.94i)31-s + ⋯
L(s)  = 1  + 0.895·3-s + (−0.368 + 0.929i)5-s + (−0.447 − 0.447i)7-s − 0.197·9-s + (−0.942 + 0.942i)11-s + (−0.330 + 0.943i)13-s + (−0.330 + 0.832i)15-s − 0.0945·17-s + (−0.870 − 0.870i)19-s + (−0.400 − 0.400i)21-s − 1.04i·23-s + (−0.727 − 0.685i)25-s − 1.07·27-s − 0.694·29-s + (1.06 + 1.06i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 - 0.408i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.912 - 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $-0.912 - 0.408i$
Analytic conductor: \(8.30444\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1040} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1040,\ (\ :1/2),\ -0.912 - 0.408i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7257657546\)
\(L(\frac12)\) \(\approx\) \(0.7257657546\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.824 - 2.07i)T \)
13 \( 1 + (1.19 - 3.40i)T \)
good3 \( 1 - 1.55T + 3T^{2} \)
7 \( 1 + (1.18 + 1.18i)T + 7iT^{2} \)
11 \( 1 + (3.12 - 3.12i)T - 11iT^{2} \)
17 \( 1 + 0.389T + 17T^{2} \)
19 \( 1 + (3.79 + 3.79i)T + 19iT^{2} \)
23 \( 1 + 4.98iT - 23T^{2} \)
29 \( 1 + 3.73T + 29T^{2} \)
31 \( 1 + (-5.94 - 5.94i)T + 31iT^{2} \)
37 \( 1 + (7.53 - 7.53i)T - 37iT^{2} \)
41 \( 1 + (-2.83 + 2.83i)T - 41iT^{2} \)
43 \( 1 - 0.482iT - 43T^{2} \)
47 \( 1 + (-9.12 - 9.12i)T + 47iT^{2} \)
53 \( 1 - 11.6iT - 53T^{2} \)
59 \( 1 + (-5.27 + 5.27i)T - 59iT^{2} \)
61 \( 1 + 6.44T + 61T^{2} \)
67 \( 1 + (-7.82 + 7.82i)T - 67iT^{2} \)
71 \( 1 + (-3.33 - 3.33i)T + 71iT^{2} \)
73 \( 1 + (7.06 - 7.06i)T - 73iT^{2} \)
79 \( 1 + 14.4iT - 79T^{2} \)
83 \( 1 + (5.83 - 5.83i)T - 83iT^{2} \)
89 \( 1 + (-8.33 - 8.33i)T + 89iT^{2} \)
97 \( 1 + (1.52 + 1.52i)T + 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.35284352924953000974479772595, −9.416512397060908367006577720028, −8.623175343759551510930748888292, −7.74032451730780979752607354550, −7.01307818787521422574142708451, −6.39354529096728223140731998616, −4.84858241167275879905152108052, −3.98049841414959549399970090184, −2.84817475643892239999186799726, −2.26679255580491913832410243194, 0.26382384993782669178558569436, 2.16863725290668930937231074257, 3.18362246080776187927933518058, 4.00885412775094185286301041091, 5.50952695832833194623738435779, 5.75226951052973205229072250018, 7.42607502915912085821887048845, 8.162710973798069856380448762323, 8.568658936270567087415508398987, 9.382406558928195436460259634331

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