Properties

Label 2-640-128.93-c1-0-15
Degree $2$
Conductor $640$
Sign $0.711 - 0.702i$
Analytic cond. $5.11042$
Root an. cond. $2.26062$
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  + (−1.12 − 0.859i)2-s + (0.972 + 0.0958i)3-s + (0.523 + 1.93i)4-s + (0.471 + 0.881i)5-s + (−1.01 − 0.943i)6-s + (0.372 + 1.87i)7-s + (1.07 − 2.61i)8-s + (−2.00 − 0.398i)9-s + (0.228 − 1.39i)10-s + (2.30 + 1.89i)11-s + (0.324 + 1.92i)12-s + (−0.360 − 0.192i)13-s + (1.18 − 2.42i)14-s + (0.374 + 0.903i)15-s + (−3.45 + 2.02i)16-s + (−1.23 + 2.98i)17-s + ⋯
L(s)  = 1  + (−0.794 − 0.607i)2-s + (0.561 + 0.0553i)3-s + (0.261 + 0.965i)4-s + (0.210 + 0.394i)5-s + (−0.412 − 0.385i)6-s + (0.140 + 0.706i)7-s + (0.378 − 0.925i)8-s + (−0.668 − 0.132i)9-s + (0.0722 − 0.441i)10-s + (0.695 + 0.570i)11-s + (0.0935 + 0.556i)12-s + (−0.0999 − 0.0534i)13-s + (0.317 − 0.646i)14-s + (0.0965 + 0.233i)15-s + (−0.863 + 0.505i)16-s + (−0.299 + 0.724i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $0.711 - 0.702i$
Analytic conductor: \(5.11042\)
Root analytic conductor: \(2.26062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 640,\ (\ :1/2),\ 0.711 - 0.702i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08138 + 0.444081i\)
\(L(\frac12)\) \(\approx\) \(1.08138 + 0.444081i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.12 + 0.859i)T \)
5 \( 1 + (-0.471 - 0.881i)T \)
good3 \( 1 + (-0.972 - 0.0958i)T + (2.94 + 0.585i)T^{2} \)
7 \( 1 + (-0.372 - 1.87i)T + (-6.46 + 2.67i)T^{2} \)
11 \( 1 + (-2.30 - 1.89i)T + (2.14 + 10.7i)T^{2} \)
13 \( 1 + (0.360 + 0.192i)T + (7.22 + 10.8i)T^{2} \)
17 \( 1 + (1.23 - 2.98i)T + (-12.0 - 12.0i)T^{2} \)
19 \( 1 + (-4.79 + 1.45i)T + (15.7 - 10.5i)T^{2} \)
23 \( 1 + (-1.24 + 0.830i)T + (8.80 - 21.2i)T^{2} \)
29 \( 1 + (-4.42 - 5.38i)T + (-5.65 + 28.4i)T^{2} \)
31 \( 1 + (6.83 - 6.83i)T - 31iT^{2} \)
37 \( 1 + (1.98 - 6.54i)T + (-30.7 - 20.5i)T^{2} \)
41 \( 1 + (-2.34 - 3.50i)T + (-15.6 + 37.8i)T^{2} \)
43 \( 1 + (7.42 - 0.731i)T + (42.1 - 8.38i)T^{2} \)
47 \( 1 + (-2.41 - 1.00i)T + (33.2 + 33.2i)T^{2} \)
53 \( 1 + (-8.24 + 10.0i)T + (-10.3 - 51.9i)T^{2} \)
59 \( 1 + (-2.43 + 1.30i)T + (32.7 - 49.0i)T^{2} \)
61 \( 1 + (0.913 - 9.27i)T + (-59.8 - 11.9i)T^{2} \)
67 \( 1 + (-0.224 + 2.28i)T + (-65.7 - 13.0i)T^{2} \)
71 \( 1 + (-6.83 + 1.35i)T + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (-2.17 + 10.9i)T + (-67.4 - 27.9i)T^{2} \)
79 \( 1 + (7.90 - 3.27i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (0.306 + 1.00i)T + (-69.0 + 46.1i)T^{2} \)
89 \( 1 + (-7.16 - 4.78i)T + (34.0 + 82.2i)T^{2} \)
97 \( 1 + (-5.48 + 5.48i)T - 97iT^{2} \)
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

−10.55684586742096588522705165297, −9.726447978968242696169776126664, −8.885031031961255609303974268387, −8.497267599920202080760795814456, −7.30700509834646598893762510963, −6.49281824070853572117194946144, −5.11758966275958843169369421157, −3.59646129479829915409482140132, −2.79566269919072277408829301802, −1.65724802532202343095813287904, 0.802618200004862330576715210584, 2.32170898522899451303268563970, 3.84256317921462969562616683134, 5.22952470900999419578600981531, 6.03588408018648565287954022535, 7.23301948509765892380842861352, 7.82559494299623819205349702258, 8.848249319798249436580366084583, 9.280396764345357606651349584549, 10.20690465726890589351009537524

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