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Properties

Label 2-2640-5.4-c1-0-9
Degree 22
Conductor 26402640
Sign 0.749−0.662i0.749 - 0.662i
Analytic cond. 21.080521.0805
Root an. cond. 4.591354.59135
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·3-s + (−1.48 − 1.67i)5-s − 2.80i·7-s − 9-s + 11-s + 5.11i·13-s + (−1.67 + 1.48i)15-s + 4.54i·17-s − 4.57·19-s − 2.80·21-s + 4i·23-s + (−0.612 + 4.96i)25-s + i·27-s + 2.38·29-s + 0.962·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (−0.662 − 0.749i)5-s − 1.06i·7-s − 0.333·9-s + 0.301·11-s + 1.41i·13-s + (−0.432 + 0.382i)15-s + 1.10i·17-s − 1.04·19-s − 0.612·21-s + 0.834i·23-s + (−0.122 + 0.992i)25-s + 0.192i·27-s + 0.443·29-s + 0.172·31-s + ⋯

Functional equation

Λ(s)=(2640s/2 ΓC(s) L(s)=((0.749−0.662i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2640s/2 ΓC(s+1/2) L(s)=((0.749−0.662i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 26402640    =    24⋅3⋅5⋅112^{4} \cdot 3 \cdot 5 \cdot 11
Sign: 0.749−0.662i0.749 - 0.662i
Analytic conductor: 21.080521.0805
Root analytic conductor: 4.591354.59135
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2640(529,⋅)\chi_{2640} (529, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2640, ( :1/2), 0.749−0.662i)(2,\ 2640,\ (\ :1/2),\ 0.749 - 0.662i)

Particular Values

L(1)L(1) ≈\approx 0.93040040440.9304004044
L(12)L(\frac12) ≈\approx 0.93040040440.9304004044
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
3 1+iT 1 + iT
5 1+(1.48+1.67i)T 1 + (1.48 + 1.67i)T
11 1−T 1 - T
good7 1+2.80iT−7T2 1 + 2.80iT - 7T^{2}
13 1−5.11iT−13T2 1 - 5.11iT - 13T^{2}
17 1−4.54iT−17T2 1 - 4.54iT - 17T^{2}
19 1+4.57T+19T2 1 + 4.57T + 19T^{2}
23 1−4iT−23T2 1 - 4iT - 23T^{2}
29 1−2.38T+29T2 1 - 2.38T + 29T^{2}
31 1−0.962T+31T2 1 - 0.962T + 31T^{2}
37 1−1.61iT−37T2 1 - 1.61iT - 37T^{2}
41 1+2.38T+41T2 1 + 2.38T + 41T^{2}
43 1−2.80iT−43T2 1 - 2.80iT - 43T^{2}
47 1−4.31iT−47T2 1 - 4.31iT - 47T^{2}
53 1−6.57iT−53T2 1 - 6.57iT - 53T^{2}
59 1+13.2T+59T2 1 + 13.2T + 59T^{2}
61 1−7.92T+61T2 1 - 7.92T + 61T^{2}
67 1+10.7iT−67T2 1 + 10.7iT - 67T^{2}
71 1−7.35T+71T2 1 - 7.35T + 71T^{2}
73 1+6.41iT−73T2 1 + 6.41iT - 73T^{2}
79 1−1.35T+79T2 1 - 1.35T + 79T^{2}
83 1+0.806iT−83T2 1 + 0.806iT - 83T^{2}
89 1−2.96T+89T2 1 - 2.96T + 89T^{2}
97 1+9.92iT−97T2 1 + 9.92iT - 97T^{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

−8.859718017805975877641640756416, −8.098054426196931291074192773546, −7.51026965223923202700426025492, −6.68522787919024080281659035753, −6.11381695950186321501888067999, −4.79082113351644864334225970656, −4.19774384854639048745462260440, −3.50543622828446913261807195992, −1.91266082906115039153081005636, −1.10955926487437224064424706370, 0.34006363423594707186063484846, 2.40387526881758131931601682775, 2.97215898817102099861262058599, 3.90567108098387751749639048552, 4.85787927526018062199198510015, 5.62647482652018955337913239359, 6.45752506735885074438415150597, 7.22951611552551668872662720916, 8.297680224396856237514722408505, 8.557266865656329546832632906577

Graph of the ZZ-function along the critical line