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Properties

Label 2-640-16.5-c1-0-7
Degree $2$
Conductor $640$
Sign $0.878 - 0.477i$
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  + (0.120 − 0.120i)3-s + (0.707 + 0.707i)5-s − 2.66i·7-s + 2.97i·9-s + (3.49 + 3.49i)11-s + (−2.94 + 2.94i)13-s + 0.169·15-s + 1.85·17-s + (3.44 − 3.44i)19-s + (−0.320 − 0.320i)21-s + 0.707i·23-s + 1.00i·25-s + (0.716 + 0.716i)27-s + (3.49 − 3.49i)29-s + 6.84·31-s + ⋯
L(s)  = 1  + (0.0692 − 0.0692i)3-s + (0.316 + 0.316i)5-s − 1.00i·7-s + 0.990i·9-s + (1.05 + 1.05i)11-s + (−0.815 + 0.815i)13-s + 0.0438·15-s + 0.448·17-s + (0.791 − 0.791i)19-s + (−0.0698 − 0.0698i)21-s + 0.147i·23-s + 0.200i·25-s + (0.137 + 0.137i)27-s + (0.649 − 0.649i)29-s + 1.22·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.878 - 0.477i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.878 - 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $0.878 - 0.477i$
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: \((2,\ 640,\ (\ :1/2),\ 0.878 - 0.477i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61534 + 0.410880i\)
\(L(\frac12)\) \(\approx\) \(1.61534 + 0.410880i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.120 + 0.120i)T - 3iT^{2} \)
7 \( 1 + 2.66iT - 7T^{2} \)
11 \( 1 + (-3.49 - 3.49i)T + 11iT^{2} \)
13 \( 1 + (2.94 - 2.94i)T - 13iT^{2} \)
17 \( 1 - 1.85T + 17T^{2} \)
19 \( 1 + (-3.44 + 3.44i)T - 19iT^{2} \)
23 \( 1 - 0.707iT - 23T^{2} \)
29 \( 1 + (-3.49 + 3.49i)T - 29iT^{2} \)
31 \( 1 - 6.84T + 31T^{2} \)
37 \( 1 + (-0.0975 - 0.0975i)T + 37iT^{2} \)
41 \( 1 - 10.2iT - 41T^{2} \)
43 \( 1 + (4.43 + 4.43i)T + 43iT^{2} \)
47 \( 1 + 1.89T + 47T^{2} \)
53 \( 1 + (-7.43 - 7.43i)T + 53iT^{2} \)
59 \( 1 + (0.959 + 0.959i)T + 59iT^{2} \)
61 \( 1 + (6.49 - 6.49i)T - 61iT^{2} \)
67 \( 1 + (3.49 - 3.49i)T - 67iT^{2} \)
71 \( 1 + 7.86iT - 71T^{2} \)
73 \( 1 + 15.6iT - 73T^{2} \)
79 \( 1 + 6.70T + 79T^{2} \)
83 \( 1 + (-3.87 + 3.87i)T - 83iT^{2} \)
89 \( 1 + 10.5iT - 89T^{2} \)
97 \( 1 - 4.79T + 97T^{2} \)
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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

−10.41475737818282266783881404917, −9.917971768152596885163670008860, −9.086793650012414200396557016273, −7.72779445440699350372770226099, −7.18203858333800096911392562378, −6.38763720388997050613092777549, −4.87960389777786518138283114006, −4.28504090550333243845661123341, −2.76597401016716425837457780022, −1.48782101026264934274981288979, 1.06435891397457918388335596630, 2.79216940042220093883235971707, 3.72630509311624703233792704400, 5.21665010960341066415357972012, 5.90572771354779047353834187730, 6.77211232863474767092407560107, 8.138539823419838654766101519881, 8.813832724020219844956900489029, 9.566379230467781908680470645574, 10.29428528454004817976767014602

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