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Properties

Label 2-640-40.3-c1-0-23
Degree $2$
Conductor $640$
Sign $0.229 - 0.973i$
Analytic cond. $5.11042$
Root an. cond. $2.26062$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 − 2i)3-s + (−2 − i)5-s + (−2 − 2i)7-s + 5i·9-s − 4·11-s + (3 − 3i)13-s + (2 + 6i)15-s + (−3 + 3i)17-s + 8i·21-s + (6 − 6i)23-s + (3 + 4i)25-s + (4 − 4i)27-s − 2·29-s + 4i·31-s + (8 + 8i)33-s + ⋯
L(s)  = 1  + (−1.15 − 1.15i)3-s + (−0.894 − 0.447i)5-s + (−0.755 − 0.755i)7-s + 1.66i·9-s − 1.20·11-s + (0.832 − 0.832i)13-s + (0.516 + 1.54i)15-s + (−0.727 + 0.727i)17-s + 1.74i·21-s + (1.25 − 1.25i)23-s + (0.600 + 0.800i)25-s + (0.769 − 0.769i)27-s − 0.371·29-s + 0.718i·31-s + (1.39 + 1.39i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $0.229 - 0.973i$
Analytic conductor: \(5.11042\)
Root analytic conductor: \(2.26062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 640,\ (\ :1/2),\ 0.229 - 0.973i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 + i)T \)
good3 \( 1 + (2 + 2i)T + 3iT^{2} \)
7 \( 1 + (2 + 2i)T + 7iT^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + (-3 + 3i)T - 13iT^{2} \)
17 \( 1 + (3 - 3i)T - 17iT^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + (-6 + 6i)T - 23iT^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 - 4iT - 31T^{2} \)
37 \( 1 + (-3 - 3i)T + 37iT^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + (-6 - 6i)T + 43iT^{2} \)
47 \( 1 + (6 + 6i)T + 47iT^{2} \)
53 \( 1 + (3 - 3i)T - 53iT^{2} \)
59 \( 1 - 8iT - 59T^{2} \)
61 \( 1 + 6iT - 61T^{2} \)
67 \( 1 + (6 - 6i)T - 67iT^{2} \)
71 \( 1 - 12iT - 71T^{2} \)
73 \( 1 + (5 + 5i)T + 73iT^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + (6 + 6i)T + 83iT^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + (-11 + 11i)T - 97iT^{2} \)
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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

−10.37827154693147119374618047683, −8.716080269415606552805172520732, −7.932016416032224379560956265380, −7.13885330676611062736412675378, −6.40752833035295989276076819704, −5.43798738630811447986049713734, −4.40967537547889203218899388139, −3.01018158300185929880400611129, −1.07238195120230068737244481521, 0, 2.85613359519106576037039529113, 3.89381254674362611964359029669, 4.86805309074938886061555059625, 5.74338258133486128520737434358, 6.60871617935077736050584726147, 7.64604549490735914164621582145, 9.011162752102313393214350662114, 9.541604600396199050328214910754, 10.65404018383037689533675362148

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