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Properties

Label 2-640-40.19-c2-0-30
Degree $2$
Conductor $640$
Sign $0.623 + 0.782i$
Analytic cond. $17.4387$
Root an. cond. $4.17597$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.62i·3-s + (4.96 + 0.561i)5-s + 6.45·7-s − 4.12·9-s − 3.94·11-s + 15.5·13-s + (2.03 − 17.9i)15-s + 25.4i·17-s + 32.0·19-s − 23.3i·21-s − 15.2·23-s + (24.3 + 5.57i)25-s − 17.6i·27-s + 34.7i·29-s − 20.2i·31-s + ⋯
L(s)  = 1  − 1.20i·3-s + (0.993 + 0.112i)5-s + 0.921·7-s − 0.458·9-s − 0.358·11-s + 1.19·13-s + (0.135 − 1.19i)15-s + 1.49i·17-s + 1.68·19-s − 1.11i·21-s − 0.664·23-s + (0.974 + 0.223i)25-s − 0.654i·27-s + 1.19i·29-s − 0.652i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.623 + 0.782i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.623 + 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $0.623 + 0.782i$
Analytic conductor: \(17.4387\)
Root analytic conductor: \(4.17597\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 640,\ (\ :1),\ 0.623 + 0.782i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.631710226\)
\(L(\frac12)\) \(\approx\) \(2.631710226\)
\(L(2)\) 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 \( 1 + (-4.96 - 0.561i)T \)
good3 \( 1 + 3.62iT - 9T^{2} \)
7 \( 1 - 6.45T + 49T^{2} \)
11 \( 1 + 3.94T + 121T^{2} \)
13 \( 1 - 15.5T + 169T^{2} \)
17 \( 1 - 25.4iT - 289T^{2} \)
19 \( 1 - 32.0T + 361T^{2} \)
23 \( 1 + 15.2T + 529T^{2} \)
29 \( 1 - 34.7iT - 841T^{2} \)
31 \( 1 + 20.2iT - 961T^{2} \)
37 \( 1 - 9.93T + 1.36e3T^{2} \)
41 \( 1 + 42.3T + 1.68e3T^{2} \)
43 \( 1 - 52.9iT - 1.84e3T^{2} \)
47 \( 1 + 21.6T + 2.20e3T^{2} \)
53 \( 1 + 35.3T + 2.80e3T^{2} \)
59 \( 1 - 16.2T + 3.48e3T^{2} \)
61 \( 1 + 77.3iT - 3.72e3T^{2} \)
67 \( 1 + 14.0iT - 4.48e3T^{2} \)
71 \( 1 + 107. iT - 5.04e3T^{2} \)
73 \( 1 + 93.7iT - 5.32e3T^{2} \)
79 \( 1 + 92.2iT - 6.24e3T^{2} \)
83 \( 1 + 26.7iT - 6.88e3T^{2} \)
89 \( 1 - 27.7T + 7.92e3T^{2} \)
97 \( 1 - 133. iT - 9.40e3T^{2} \)
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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

−10.35841812442565693126423952518, −9.314122089729279219250946185099, −8.212561973633702106793616980572, −7.74557260552413161831369272199, −6.54233845968952220809647749730, −5.95438699323570711144711990907, −4.93488263616357547001381824110, −3.33956836131139554382435995676, −1.82341472211595572263461786121, −1.32310947604840868830752181526, 1.29961768260558215841506570154, 2.80782463375935951558936869353, 4.03213200365735979198045127307, 5.14785679519744224110050748236, 5.50125085241546627932468108392, 6.89688835769636064494629076951, 8.070498646315403968396342434069, 8.995235472822060929496752753203, 9.745409799594894365619195185908, 10.28327024045613521265087294725

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