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Properties

Label 2-640-40.27-c1-0-12
Degree $2$
Conductor $640$
Sign $0.973 - 0.229i$
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  + (2 − i)5-s + 3i·9-s + (5 + 5i)13-s + (−3 − 3i)17-s + (3 − 4i)25-s + 10·29-s + (5 − 5i)37-s − 8·41-s + (3 + 6i)45-s + 7i·49-s + (−5 − 5i)53-s + 10i·61-s + (15 + 5i)65-s + (11 − 11i)73-s − 9·81-s + ⋯
L(s)  = 1  + (0.894 − 0.447i)5-s + i·9-s + (1.38 + 1.38i)13-s + (−0.727 − 0.727i)17-s + (0.600 − 0.800i)25-s + 1.85·29-s + (0.821 − 0.821i)37-s − 1.24·41-s + (0.447 + 0.894i)45-s + i·49-s + (−0.686 − 0.686i)53-s + 1.28i·61-s + (1.86 + 0.620i)65-s + (1.28 − 1.28i)73-s − 81-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $0.973 - 0.229i$
Analytic conductor: \(5.11042\)
Root analytic conductor: \(2.26062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (447, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 640,\ (\ :1/2),\ 0.973 - 0.229i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.79097 + 0.208530i\)
\(L(\frac12)\) \(\approx\) \(1.79097 + 0.208530i\)
\(L(\frac{3}{2})\) 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 \( 1 - 3iT^{2} \)
7 \( 1 - 7iT^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + (-5 - 5i)T + 13iT^{2} \)
17 \( 1 + (3 + 3i)T + 17iT^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 23iT^{2} \)
29 \( 1 - 10T + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + (-5 + 5i)T - 37iT^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 - 47iT^{2} \)
53 \( 1 + (5 + 5i)T + 53iT^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 - 10iT - 61T^{2} \)
67 \( 1 + 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-11 + 11i)T - 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 - 16iT - 89T^{2} \)
97 \( 1 + (13 + 13i)T + 97iT^{2} \)
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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

−10.65767338927181772321516981806, −9.656067500408260093938161875664, −8.884489111685687606360383364266, −8.208456564630146961974742584140, −6.86542742522800213775067869616, −6.16604991613569018299740908008, −5.04073816145963120921586494585, −4.25650204458828518379193223841, −2.58447357404155016141704038582, −1.47774598846936886897286320391, 1.20775996873364005314871711799, 2.81448259458608918789009487815, 3.74490299329917498493524187894, 5.19857404185808768269645006621, 6.31517888356130832358685866429, 6.54182879850666127986144728761, 8.116407126842502297632691688064, 8.766128671741601924661907876758, 9.816763381046881515957460382662, 10.47327371233460928634585879200

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