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Properties

Label 2-1350-15.8-c1-0-18
Degree 22
Conductor 13501350
Sign −0.437+0.899i-0.437 + 0.899i
Analytic cond. 10.779810.7798
Root an. cond. 3.283263.28326
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  + (−0.707 − 0.707i)2-s + 1.00i·4-s + (1.22 − 1.22i)7-s + (0.707 − 0.707i)8-s − 1.73·14-s − 1.00·16-s + (−4.24 − 4.24i)17-s + i·19-s + (4.24 − 4.24i)23-s + (1.22 + 1.22i)28-s − 10.3·29-s + 7·31-s + (0.707 + 0.707i)32-s + 6i·34-s + (6.12 − 6.12i)37-s + (0.707 − 0.707i)38-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.500i·4-s + (0.462 − 0.462i)7-s + (0.250 − 0.250i)8-s − 0.462·14-s − 0.250·16-s + (−1.02 − 1.02i)17-s + 0.229i·19-s + (0.884 − 0.884i)23-s + (0.231 + 0.231i)28-s − 1.92·29-s + 1.25·31-s + (0.125 + 0.125i)32-s + 1.02i·34-s + (1.00 − 1.00i)37-s + (0.114 − 0.114i)38-s + ⋯

Functional equation

Λ(s)=(1350s/2 ΓC(s) L(s)=((−0.437+0.899i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.437 + 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1350s/2 ΓC(s+1/2) L(s)=((−0.437+0.899i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13501350    =    2⋅33⋅522 \cdot 3^{3} \cdot 5^{2}
Sign: −0.437+0.899i-0.437 + 0.899i
Analytic conductor: 10.779810.7798
Root analytic conductor: 3.283263.28326
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1350(593,⋅)\chi_{1350} (593, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1350, ( :1/2), −0.437+0.899i)(2,\ 1350,\ (\ :1/2),\ -0.437 + 0.899i)

Particular Values

L(1)L(1) ≈\approx 1.0442652001.044265200
L(12)L(\frac12) ≈\approx 1.0442652001.044265200
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+(0.707+0.707i)T 1 + (0.707 + 0.707i)T
3 1 1
5 1 1
good7 1+(−1.22+1.22i)T−7iT2 1 + (-1.22 + 1.22i)T - 7iT^{2}
11 1−11T2 1 - 11T^{2}
13 1+13iT2 1 + 13iT^{2}
17 1+(4.24+4.24i)T+17iT2 1 + (4.24 + 4.24i)T + 17iT^{2}
19 1−iT−19T2 1 - iT - 19T^{2}
23 1+(−4.24+4.24i)T−23iT2 1 + (-4.24 + 4.24i)T - 23iT^{2}
29 1+10.3T+29T2 1 + 10.3T + 29T^{2}
31 1−7T+31T2 1 - 7T + 31T^{2}
37 1+(−6.12+6.12i)T−37iT2 1 + (-6.12 + 6.12i)T - 37iT^{2}
41 1−41T2 1 - 41T^{2}
43 1+(1.22+1.22i)T+43iT2 1 + (1.22 + 1.22i)T + 43iT^{2}
47 1+47iT2 1 + 47iT^{2}
53 1+(−8.48+8.48i)T−53iT2 1 + (-8.48 + 8.48i)T - 53iT^{2}
59 1+10.3T+59T2 1 + 10.3T + 59T^{2}
61 1−5T+61T2 1 - 5T + 61T^{2}
67 1+(7.34−7.34i)T−67iT2 1 + (7.34 - 7.34i)T - 67iT^{2}
71 1+10.3iT−71T2 1 + 10.3iT - 71T^{2}
73 1+(8.57+8.57i)T+73iT2 1 + (8.57 + 8.57i)T + 73iT^{2}
79 1+13iT−79T2 1 + 13iT - 79T^{2}
83 1+(−4.24+4.24i)T−83iT2 1 + (-4.24 + 4.24i)T - 83iT^{2}
89 1−10.3T+89T2 1 - 10.3T + 89T^{2}
97 1+(−6.12+6.12i)T−97iT2 1 + (-6.12 + 6.12i)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

−9.265976322787672410733787142119, −8.764304954639611809708428761377, −7.70581614263989836076679329103, −7.17324719902625095893494538193, −6.14045086861577677841951339377, −4.88956214321609264556287449138, −4.17806872238576729423685088755, −2.97404724489201598206639185458, −1.94094925174734476547812087214, −0.53038890879752740608558706836, 1.38822748557989546825409967720, 2.55043497088047339196635734269, 3.98166576338394932210969768525, 4.98142418280382511875373978787, 5.82867178915113534617917488329, 6.63014424433102306854535550566, 7.52826539603289712069000422792, 8.288518530224666217556067836335, 8.981745418593224473826828974507, 9.642730809841484461149969709354

Graph of the ZZ-function along the critical line