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Properties

Label 2-640-128.93-c1-0-26
Degree $2$
Conductor $640$
Sign $0.911 - 0.411i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.652 + 1.25i)2-s + (−0.645 − 0.0635i)3-s + (−1.14 − 1.63i)4-s + (0.471 + 0.881i)5-s + (0.500 − 0.768i)6-s + (−0.153 − 0.769i)7-s + (2.80 − 0.374i)8-s + (−2.53 − 0.503i)9-s + (−1.41 + 0.0163i)10-s + (−1.07 − 0.881i)11-s + (0.637 + 1.12i)12-s + (1.53 + 0.818i)13-s + (1.06 + 0.309i)14-s + (−0.248 − 0.599i)15-s + (−1.35 + 3.76i)16-s + (−0.236 + 0.570i)17-s + ⋯
L(s)  = 1  + (−0.461 + 0.887i)2-s + (−0.372 − 0.0366i)3-s + (−0.574 − 0.818i)4-s + (0.210 + 0.394i)5-s + (0.204 − 0.313i)6-s + (−0.0578 − 0.290i)7-s + (0.991 − 0.132i)8-s + (−0.843 − 0.167i)9-s + (−0.447 + 0.00515i)10-s + (−0.323 − 0.265i)11-s + (0.184 + 0.325i)12-s + (0.424 + 0.226i)13-s + (0.284 + 0.0828i)14-s + (−0.0640 − 0.154i)15-s + (−0.339 + 0.940i)16-s + (−0.0572 + 0.138i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $0.911 - 0.411i$
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: \((2,\ 640,\ (\ :1/2),\ 0.911 - 0.411i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.907474 + 0.195478i\)
\(L(\frac12)\) \(\approx\) \(0.907474 + 0.195478i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.652 - 1.25i)T \)
5 \( 1 + (-0.471 - 0.881i)T \)
good3 \( 1 + (0.645 + 0.0635i)T + (2.94 + 0.585i)T^{2} \)
7 \( 1 + (0.153 + 0.769i)T + (-6.46 + 2.67i)T^{2} \)
11 \( 1 + (1.07 + 0.881i)T + (2.14 + 10.7i)T^{2} \)
13 \( 1 + (-1.53 - 0.818i)T + (7.22 + 10.8i)T^{2} \)
17 \( 1 + (0.236 - 0.570i)T + (-12.0 - 12.0i)T^{2} \)
19 \( 1 + (-7.87 + 2.39i)T + (15.7 - 10.5i)T^{2} \)
23 \( 1 + (-4.62 + 3.09i)T + (8.80 - 21.2i)T^{2} \)
29 \( 1 + (4.50 + 5.48i)T + (-5.65 + 28.4i)T^{2} \)
31 \( 1 + (0.239 - 0.239i)T - 31iT^{2} \)
37 \( 1 + (-2.34 + 7.73i)T + (-30.7 - 20.5i)T^{2} \)
41 \( 1 + (-5.57 - 8.34i)T + (-15.6 + 37.8i)T^{2} \)
43 \( 1 + (-7.75 + 0.763i)T + (42.1 - 8.38i)T^{2} \)
47 \( 1 + (-12.0 - 4.98i)T + (33.2 + 33.2i)T^{2} \)
53 \( 1 + (2.02 - 2.47i)T + (-10.3 - 51.9i)T^{2} \)
59 \( 1 + (12.3 - 6.58i)T + (32.7 - 49.0i)T^{2} \)
61 \( 1 + (1.30 - 13.2i)T + (-59.8 - 11.9i)T^{2} \)
67 \( 1 + (-0.883 + 8.97i)T + (-65.7 - 13.0i)T^{2} \)
71 \( 1 + (-1.87 + 0.373i)T + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (-2.28 + 11.4i)T + (-67.4 - 27.9i)T^{2} \)
79 \( 1 + (-8.21 + 3.40i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (0.336 + 1.10i)T + (-69.0 + 46.1i)T^{2} \)
89 \( 1 + (7.63 + 5.10i)T + (34.0 + 82.2i)T^{2} \)
97 \( 1 + (3.11 - 3.11i)T - 97iT^{2} \)
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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

−10.83735054755214502238757945036, −9.439978450566634627499124115059, −9.038250214513637578722270381916, −7.78832251717426831406536021691, −7.17970157827796469948923479396, −6.03070041189360102196447816153, −5.60900333886962605236685276848, −4.34276782828719417199596902743, −2.83139845869626742287376918826, −0.809512599484262944766571536226, 1.08451844899555543789222121117, 2.60478735881725846254350753035, 3.62621401818700821629756007350, 5.10364849027018047529309497350, 5.63562870721506544856594370505, 7.24071819920960249274946254582, 8.079514881003499673486960235456, 9.072272078467355353019667559910, 9.556109966044843360906500562454, 10.70180545699792961580793213360

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