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Properties

Label 2-640-16.13-c1-0-15
Degree $2$
Conductor $640$
Sign $-0.992 + 0.122i$
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

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

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $-0.992 + 0.122i$
Analytic conductor: \(5.11042\)
Root analytic conductor: \(2.26062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 640,\ (\ :1/2),\ -0.992 + 0.122i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0544136 - 0.884223i\)
\(L(\frac12)\) \(\approx\) \(0.0544136 - 0.884223i\)
\(L(\frac{3}{2})\) 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

−10.32144461494856875599062285846, −9.491278745310807865495127251381, −8.003837553439470135583211889900, −7.42037374745875238912405695045, −6.57122714215371961181585652905, −5.79548866557724403348486243001, −4.74931926166660381498975160771, −3.51031489249518269839657256711, −1.52994293029383008565276429149, −0.57427552089375237632694646787, 2.11217082589990067926680615793, 3.50221417269710858600894623632, 4.86297958373482484616052528922, 5.43424628240184697048895211346, 6.20827447766950129584503380297, 7.26292543864577757597604673364, 8.780575675256253005996165529764, 9.451065851608612067248137409928, 10.01499695840955116223341124842, 11.07624437704541961591106234289

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