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Properties

Label 2-640-128.85-c1-0-9
Degree $2$
Conductor $640$
Sign $-0.928 - 0.371i$
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  + (−0.106 + 1.41i)2-s + (0.661 − 0.806i)3-s + (−1.97 − 0.300i)4-s + (−0.956 + 0.290i)5-s + (1.06 + 1.01i)6-s + (0.522 + 0.103i)7-s + (0.634 − 2.75i)8-s + (0.373 + 1.87i)9-s + (−0.307 − 1.38i)10-s + (−3.97 − 0.391i)11-s + (−1.55 + 1.39i)12-s + (−0.214 + 0.707i)13-s + (−0.202 + 0.725i)14-s + (−0.399 + 0.963i)15-s + (3.81 + 1.18i)16-s + (2.48 + 6.00i)17-s + ⋯
L(s)  = 1  + (−0.0753 + 0.997i)2-s + (0.381 − 0.465i)3-s + (−0.988 − 0.150i)4-s + (−0.427 + 0.129i)5-s + (0.435 + 0.415i)6-s + (0.197 + 0.0392i)7-s + (0.224 − 0.974i)8-s + (0.124 + 0.625i)9-s + (−0.0971 − 0.436i)10-s + (−1.19 − 0.117i)11-s + (−0.447 + 0.402i)12-s + (−0.0595 + 0.196i)13-s + (−0.0540 + 0.193i)14-s + (−0.103 + 0.248i)15-s + (0.954 + 0.297i)16-s + (0.603 + 1.45i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.928 - 0.371i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.928 - 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $-0.928 - 0.371i$
Analytic conductor: \(5.11042\)
Root analytic conductor: \(2.26062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 640,\ (\ :1/2),\ -0.928 - 0.371i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.162290 + 0.842014i\)
\(L(\frac12)\) \(\approx\) \(0.162290 + 0.842014i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.106 - 1.41i)T \)
5 \( 1 + (0.956 - 0.290i)T \)
good3 \( 1 + (-0.661 + 0.806i)T + (-0.585 - 2.94i)T^{2} \)
7 \( 1 + (-0.522 - 0.103i)T + (6.46 + 2.67i)T^{2} \)
11 \( 1 + (3.97 + 0.391i)T + (10.7 + 2.14i)T^{2} \)
13 \( 1 + (0.214 - 0.707i)T + (-10.8 - 7.22i)T^{2} \)
17 \( 1 + (-2.48 - 6.00i)T + (-12.0 + 12.0i)T^{2} \)
19 \( 1 + (5.94 - 3.17i)T + (10.5 - 15.7i)T^{2} \)
23 \( 1 + (2.92 - 4.37i)T + (-8.80 - 21.2i)T^{2} \)
29 \( 1 + (0.280 + 2.84i)T + (-28.4 + 5.65i)T^{2} \)
31 \( 1 + (-5.31 - 5.31i)T + 31iT^{2} \)
37 \( 1 + (-0.269 + 0.504i)T + (-20.5 - 30.7i)T^{2} \)
41 \( 1 + (4.00 + 2.67i)T + (15.6 + 37.8i)T^{2} \)
43 \( 1 + (6.00 + 7.31i)T + (-8.38 + 42.1i)T^{2} \)
47 \( 1 + (4.04 - 1.67i)T + (33.2 - 33.2i)T^{2} \)
53 \( 1 + (0.849 - 8.62i)T + (-51.9 - 10.3i)T^{2} \)
59 \( 1 + (-2.40 - 7.91i)T + (-49.0 + 32.7i)T^{2} \)
61 \( 1 + (-10.4 - 8.56i)T + (11.9 + 59.8i)T^{2} \)
67 \( 1 + (9.25 + 7.59i)T + (13.0 + 65.7i)T^{2} \)
71 \( 1 + (-2.43 + 12.2i)T + (-65.5 - 27.1i)T^{2} \)
73 \( 1 + (-10.6 + 2.11i)T + (67.4 - 27.9i)T^{2} \)
79 \( 1 + (-8.63 - 3.57i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (6.99 + 13.0i)T + (-46.1 + 69.0i)T^{2} \)
89 \( 1 + (-3.41 - 5.11i)T + (-34.0 + 82.2i)T^{2} \)
97 \( 1 + (2.87 + 2.87i)T + 97iT^{2} \)
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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

−10.46120017282301417619310829579, −10.29876777796155740220692642287, −8.720884146149576440766392304803, −8.058647914283214303571871782661, −7.73280288988242484749395972953, −6.60255995957956521663105173018, −5.66016524453489939654107254974, −4.65393232873716878175204146518, −3.52515169895531975735316488007, −1.85382658294821212225032433884, 0.45143095028459011977952639452, 2.43464577947903271821635177118, 3.30525959130475186009338988388, 4.48168267669276802322446536223, 5.07869756995142388724732518485, 6.67397435871543549557797950606, 8.065534418134296786287127278266, 8.454059045358419104472221666053, 9.680560545863919341507785773939, 10.02416669461378400484423117408

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