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Properties

Label 2-640-16.5-c1-0-15
Degree $2$
Conductor $640$
Sign $-0.122 + 0.992i$
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  + (1.82 − 1.82i)3-s + (0.707 + 0.707i)5-s − 4.50i·7-s − 3.68i·9-s + (−1.64 − 1.64i)11-s + (−1.51 + 1.51i)13-s + 2.58·15-s + 1.45·17-s + (−2.67 + 2.67i)19-s + (−8.24 − 8.24i)21-s − 2.37i·23-s + 1.00i·25-s + (−1.24 − 1.24i)27-s + (−0.924 + 0.924i)29-s + 7.20·31-s + ⋯
L(s)  = 1  + (1.05 − 1.05i)3-s + (0.316 + 0.316i)5-s − 1.70i·7-s − 1.22i·9-s + (−0.494 − 0.494i)11-s + (−0.421 + 0.421i)13-s + 0.667·15-s + 0.353·17-s + (−0.614 + 0.614i)19-s + (−1.79 − 1.79i)21-s − 0.495i·23-s + 0.200i·25-s + (−0.239 − 0.239i)27-s + (−0.171 + 0.171i)29-s + 1.29·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.122 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.122 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $-0.122 + 0.992i$
Analytic conductor: \(5.11042\)
Root analytic conductor: \(2.26062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 640,\ (\ :1/2),\ -0.122 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37164 - 1.55153i\)
\(L(\frac12)\) \(\approx\) \(1.37164 - 1.55153i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (-1.82 + 1.82i)T - 3iT^{2} \)
7 \( 1 + 4.50iT - 7T^{2} \)
11 \( 1 + (1.64 + 1.64i)T + 11iT^{2} \)
13 \( 1 + (1.51 - 1.51i)T - 13iT^{2} \)
17 \( 1 - 1.45T + 17T^{2} \)
19 \( 1 + (2.67 - 2.67i)T - 19iT^{2} \)
23 \( 1 + 2.37iT - 23T^{2} \)
29 \( 1 + (0.924 - 0.924i)T - 29iT^{2} \)
31 \( 1 - 7.20T + 31T^{2} \)
37 \( 1 + (-5.21 - 5.21i)T + 37iT^{2} \)
41 \( 1 + 6.41iT - 41T^{2} \)
43 \( 1 + (-7.65 - 7.65i)T + 43iT^{2} \)
47 \( 1 - 2.51T + 47T^{2} \)
53 \( 1 + (1.50 + 1.50i)T + 53iT^{2} \)
59 \( 1 + (5.31 + 5.31i)T + 59iT^{2} \)
61 \( 1 + (-1.02 + 1.02i)T - 61iT^{2} \)
67 \( 1 + (-5.22 + 5.22i)T - 67iT^{2} \)
71 \( 1 - 1.92iT - 71T^{2} \)
73 \( 1 + 1.39iT - 73T^{2} \)
79 \( 1 + 5.06T + 79T^{2} \)
83 \( 1 + (2.44 - 2.44i)T - 83iT^{2} \)
89 \( 1 - 9.36iT - 89T^{2} \)
97 \( 1 - 18.6T + 97T^{2} \)
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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

−10.32475913691423100818947763671, −9.473469622520775047948288344550, −8.250900288661293274852491597735, −7.75759790918387796137304932240, −6.95947695258251717031604086856, −6.21984325298693990365255809409, −4.55737117604517642817524670146, −3.43858438081427747764553315276, −2.37992413019381972104989850182, −1.05176549287948235481567258361, 2.31553219028991107923101858604, 2.88221502157867078071753074321, 4.32890509980764508660372507913, 5.18944298054916215798788768079, 6.04665220566187116242300433888, 7.63441281631879404453004257063, 8.503544119979257337994149765773, 9.118549249634807163322668297785, 9.715292711429134791978746188517, 10.47813919969583780587524710557

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