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Properties

Label 2-1350-15.8-c1-0-9
Degree 22
Conductor 13501350
Sign −0.525−0.850i-0.525 - 0.850i
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 + (−0.707 + 0.707i)8-s + 5.19i·11-s + (3.67 + 3.67i)13-s − 1.00·16-s + (−2.12 − 2.12i)17-s − 2i·19-s + (−3.67 + 3.67i)22-s + (2.12 − 2.12i)23-s + 5.19i·26-s − 5.19·29-s − 5·31-s + (−0.707 − 0.707i)32-s − 3i·34-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.500i·4-s + (−0.250 + 0.250i)8-s + 1.56i·11-s + (1.01 + 1.01i)13-s − 0.250·16-s + (−0.514 − 0.514i)17-s − 0.458i·19-s + (−0.783 + 0.783i)22-s + (0.442 − 0.442i)23-s + 1.01i·26-s − 0.964·29-s − 0.898·31-s + (−0.125 − 0.125i)32-s − 0.514i·34-s + ⋯

Functional equation

Λ(s)=(1350s/2 ΓC(s) L(s)=((−0.525−0.850i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1350s/2 ΓC(s+1/2) L(s)=((−0.525−0.850i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13501350    =    2⋅33⋅522 \cdot 3^{3} \cdot 5^{2}
Sign: −0.525−0.850i-0.525 - 0.850i
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), −0.525−0.850i)(2,\ 1350,\ (\ :1/2),\ -0.525 - 0.850i)

Particular Values

L(1)L(1) ≈\approx 1.9249289531.924928953
L(12)L(\frac12) ≈\approx 1.9249289531.924928953
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 1−7iT2 1 - 7iT^{2}
11 1−5.19iT−11T2 1 - 5.19iT - 11T^{2}
13 1+(−3.67−3.67i)T+13iT2 1 + (-3.67 - 3.67i)T + 13iT^{2}
17 1+(2.12+2.12i)T+17iT2 1 + (2.12 + 2.12i)T + 17iT^{2}
19 1+2iT−19T2 1 + 2iT - 19T^{2}
23 1+(−2.12+2.12i)T−23iT2 1 + (-2.12 + 2.12i)T - 23iT^{2}
29 1+5.19T+29T2 1 + 5.19T + 29T^{2}
31 1+5T+31T2 1 + 5T + 31T^{2}
37 1−37iT2 1 - 37iT^{2}
41 1−10.3iT−41T2 1 - 10.3iT - 41T^{2}
43 1+(−3.67−3.67i)T+43iT2 1 + (-3.67 - 3.67i)T + 43iT^{2}
47 1+(−6.36−6.36i)T+47iT2 1 + (-6.36 - 6.36i)T + 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 1−8T+61T2 1 - 8T + 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+(−7.34−7.34i)T+73iT2 1 + (-7.34 - 7.34i)T + 73iT^{2}
79 1+iT−79T2 1 + iT - 79T^{2}
83 1+(−8.48+8.48i)T−83iT2 1 + (-8.48 + 8.48i)T - 83iT^{2}
89 1+10.3T+89T2 1 + 10.3T + 89T^{2}
97 1+(−7.34+7.34i)T−97iT2 1 + (-7.34 + 7.34i)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

−9.453657014601553805223068767804, −9.278145431351144729795561514799, −8.107548349008261409009894857866, −7.22682532797175639858678736859, −6.69032646073399130716297855500, −5.78697010448664637321266142963, −4.60747264623717742620514149934, −4.24486351913321047497330569111, −2.88128404095386189726962634435, −1.69167693584895768397947699139, 0.67489116149060552623711061002, 2.05766479338680668056488943355, 3.52706953563382768068521759879, 3.67804464933517959546343540395, 5.34309895184805718106719920135, 5.73236598260454551736354737837, 6.65344285338557107792346012807, 7.84612751083603661047838127377, 8.623767844733135245901156125799, 9.277110441320040440214824674661

Graph of the ZZ-function along the critical line