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Properties

Label 2-640-80.67-c1-0-7
Degree $2$
Conductor $640$
Sign $0.665 - 0.746i$
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  + 0.614i·3-s + (2.07 + 0.832i)5-s + (2.83 + 2.83i)7-s + 2.62·9-s + (−1.95 − 1.95i)11-s + 2.05·13-s + (−0.511 + 1.27i)15-s + (−4.06 − 4.06i)17-s + (−0.683 − 0.683i)19-s + (−1.74 + 1.74i)21-s + (−4.95 + 4.95i)23-s + (3.61 + 3.45i)25-s + 3.45i·27-s + (0.835 − 0.835i)29-s − 2.35i·31-s + ⋯
L(s)  = 1  + 0.354i·3-s + (0.928 + 0.372i)5-s + (1.07 + 1.07i)7-s + 0.874·9-s + (−0.590 − 0.590i)11-s + 0.569·13-s + (−0.132 + 0.329i)15-s + (−0.986 − 0.986i)17-s + (−0.156 − 0.156i)19-s + (−0.380 + 0.380i)21-s + (−1.03 + 1.03i)23-s + (0.723 + 0.690i)25-s + 0.664i·27-s + (0.155 − 0.155i)29-s − 0.423i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.665 - 0.746i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.665 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $0.665 - 0.746i$
Analytic conductor: \(5.11042\)
Root analytic conductor: \(2.26062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 640,\ (\ :1/2),\ 0.665 - 0.746i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.80427 + 0.808569i\)
\(L(\frac12)\) \(\approx\) \(1.80427 + 0.808569i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.07 - 0.832i)T \)
good3 \( 1 - 0.614iT - 3T^{2} \)
7 \( 1 + (-2.83 - 2.83i)T + 7iT^{2} \)
11 \( 1 + (1.95 + 1.95i)T + 11iT^{2} \)
13 \( 1 - 2.05T + 13T^{2} \)
17 \( 1 + (4.06 + 4.06i)T + 17iT^{2} \)
19 \( 1 + (0.683 + 0.683i)T + 19iT^{2} \)
23 \( 1 + (4.95 - 4.95i)T - 23iT^{2} \)
29 \( 1 + (-0.835 + 0.835i)T - 29iT^{2} \)
31 \( 1 + 2.35iT - 31T^{2} \)
37 \( 1 - 4.54T + 37T^{2} \)
41 \( 1 + 5.07iT - 41T^{2} \)
43 \( 1 - 0.849T + 43T^{2} \)
47 \( 1 + (-2.72 + 2.72i)T - 47iT^{2} \)
53 \( 1 - 5.17iT - 53T^{2} \)
59 \( 1 + (4.16 - 4.16i)T - 59iT^{2} \)
61 \( 1 + (5.55 + 5.55i)T + 61iT^{2} \)
67 \( 1 - 1.73T + 67T^{2} \)
71 \( 1 - 2.33T + 71T^{2} \)
73 \( 1 + (-4.39 - 4.39i)T + 73iT^{2} \)
79 \( 1 + 14.0T + 79T^{2} \)
83 \( 1 + 2.75iT - 83T^{2} \)
89 \( 1 + 11.6T + 89T^{2} \)
97 \( 1 + (3.52 + 3.52i)T + 97iT^{2} \)
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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

−10.79040181189357351172439686578, −9.740065109113141503928778475339, −9.088035615015808300236128384554, −8.189603294108894076468960736704, −7.14722107984417076233017277283, −5.96092782009488585560018219745, −5.32310964534052512350183452452, −4.28470459193835896534991690937, −2.73245300745534809019943120252, −1.74368284576097245233296858990, 1.29976646088413326268112484405, 2.15538145794367224929878677283, 4.20521728455311090380248521281, 4.70456913923788876017472802550, 6.06567947724140952169351385885, 6.87972122212006103129312177420, 7.896009457987212387344684486236, 8.530957650649805085551239719734, 9.802042741162453887883635466973, 10.46065683538679948490140941442

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