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Properties

Label 2-640-40.3-c1-0-23
Degree 22
Conductor 640640
Sign 0.229−0.973i0.229 - 0.973i
Analytic cond. 5.110425.11042
Root an. cond. 2.260622.26062
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 11

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

Λ(s)=(640s/2 ΓC(s) L(s)=((0.229−0.973i) Λ‾(2−s)\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}
Λ(s)=(640s/2 ΓC(s+1/2) L(s)=((0.229−0.973i) Λ‾(1−s)\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: 22
Conductor: 640640    =    27⋅52^{7} \cdot 5
Sign: 0.229−0.973i0.229 - 0.973i
Analytic conductor: 5.110425.11042
Root analytic conductor: 2.260622.26062
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ640(63,⋅)\chi_{640} (63, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 11
Selberg data: (2, 640, ( :1/2), 0.229−0.973i)(2,\ 640,\ (\ :1/2),\ 0.229 - 0.973i)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
5 1+(2+i)T 1 + (2 + i)T
good3 1+(2+2i)T+3iT2 1 + (2 + 2i)T + 3iT^{2}
7 1+(2+2i)T+7iT2 1 + (2 + 2i)T + 7iT^{2}
11 1+4T+11T2 1 + 4T + 11T^{2}
13 1+(−3+3i)T−13iT2 1 + (-3 + 3i)T - 13iT^{2}
17 1+(3−3i)T−17iT2 1 + (3 - 3i)T - 17iT^{2}
19 1−19T2 1 - 19T^{2}
23 1+(−6+6i)T−23iT2 1 + (-6 + 6i)T - 23iT^{2}
29 1+2T+29T2 1 + 2T + 29T^{2}
31 1−4iT−31T2 1 - 4iT - 31T^{2}
37 1+(−3−3i)T+37iT2 1 + (-3 - 3i)T + 37iT^{2}
41 1+41T2 1 + 41T^{2}
43 1+(−6−6i)T+43iT2 1 + (-6 - 6i)T + 43iT^{2}
47 1+(6+6i)T+47iT2 1 + (6 + 6i)T + 47iT^{2}
53 1+(3−3i)T−53iT2 1 + (3 - 3i)T - 53iT^{2}
59 1−8iT−59T2 1 - 8iT - 59T^{2}
61 1+6iT−61T2 1 + 6iT - 61T^{2}
67 1+(6−6i)T−67iT2 1 + (6 - 6i)T - 67iT^{2}
71 1−12iT−71T2 1 - 12iT - 71T^{2}
73 1+(5+5i)T+73iT2 1 + (5 + 5i)T + 73iT^{2}
79 1+8T+79T2 1 + 8T + 79T^{2}
83 1+(6+6i)T+83iT2 1 + (6 + 6i)T + 83iT^{2}
89 1−89T2 1 - 89T^{2}
97 1+(−11+11i)T−97iT2 1 + (-11 + 11i)T - 97iT^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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 ZZ-function along the critical line