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Properties

Label 2-507-39.5-c1-0-4
Degree 22
Conductor 507507
Sign 0.0736−0.997i0.0736 - 0.997i
Analytic cond. 4.048414.04841
Root an. cond. 2.012062.01206
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  + (−0.540 − 0.540i)2-s + (0.0858 + 1.72i)3-s − 1.41i·4-s + (0.996 + 0.996i)5-s + (0.888 − 0.981i)6-s + (1.80 + 1.80i)7-s + (−1.84 + 1.84i)8-s + (−2.98 + 0.296i)9-s − 1.07i·10-s + (−3.35 + 3.35i)11-s + (2.44 − 0.121i)12-s − 1.94i·14-s + (−1.63 + 1.80i)15-s − 0.837·16-s + 5.80·17-s + (1.77 + 1.45i)18-s + ⋯
L(s)  = 1  + (−0.382 − 0.382i)2-s + (0.0495 + 0.998i)3-s − 0.708i·4-s + (0.445 + 0.445i)5-s + (0.362 − 0.400i)6-s + (0.681 + 0.681i)7-s + (−0.652 + 0.652i)8-s + (−0.995 + 0.0989i)9-s − 0.340i·10-s + (−1.01 + 1.01i)11-s + (0.707 − 0.0350i)12-s − 0.520i·14-s + (−0.422 + 0.467i)15-s − 0.209·16-s + 1.40·17-s + (0.417 + 0.342i)18-s + ⋯

Functional equation

Λ(s)=(507s/2 ΓC(s) L(s)=((0.0736−0.997i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0736 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(507s/2 ΓC(s+1/2) L(s)=((0.0736−0.997i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0736 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 507507    =    3⋅1323 \cdot 13^{2}
Sign: 0.0736−0.997i0.0736 - 0.997i
Analytic conductor: 4.048414.04841
Root analytic conductor: 2.012062.01206
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ507(239,⋅)\chi_{507} (239, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 507, ( :1/2), 0.0736−0.997i)(2,\ 507,\ (\ :1/2),\ 0.0736 - 0.997i)

Particular Values

L(1)L(1) ≈\approx 0.787995+0.731921i0.787995 + 0.731921i
L(12)L(\frac12) ≈\approx 0.787995+0.731921i0.787995 + 0.731921i
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)
bad3 1+(−0.0858−1.72i)T 1 + (-0.0858 - 1.72i)T
13 1 1
good2 1+(0.540+0.540i)T+2iT2 1 + (0.540 + 0.540i)T + 2iT^{2}
5 1+(−0.996−0.996i)T+5iT2 1 + (-0.996 - 0.996i)T + 5iT^{2}
7 1+(−1.80−1.80i)T+7iT2 1 + (-1.80 - 1.80i)T + 7iT^{2}
11 1+(3.35−3.35i)T−11iT2 1 + (3.35 - 3.35i)T - 11iT^{2}
17 1−5.80T+17T2 1 - 5.80T + 17T^{2}
19 1+(2.39−2.39i)T−19iT2 1 + (2.39 - 2.39i)T - 19iT^{2}
23 1+3.39T+23T2 1 + 3.39T + 23T^{2}
29 1−6.57iT−29T2 1 - 6.57iT - 29T^{2}
31 1+(0.386−0.386i)T−31iT2 1 + (0.386 - 0.386i)T - 31iT^{2}
37 1+(−5.93−5.93i)T+37iT2 1 + (-5.93 - 5.93i)T + 37iT^{2}
41 1+(0.734+0.734i)T+41iT2 1 + (0.734 + 0.734i)T + 41iT^{2}
43 1−7.56iT−43T2 1 - 7.56iT - 43T^{2}
47 1+(−0.243+0.243i)T−47iT2 1 + (-0.243 + 0.243i)T - 47iT^{2}
53 1−2.07iT−53T2 1 - 2.07iT - 53T^{2}
59 1+(3.56−3.56i)T−59iT2 1 + (3.56 - 3.56i)T - 59iT^{2}
61 1−7.04T+61T2 1 - 7.04T + 61T^{2}
67 1+(−4.54+4.54i)T−67iT2 1 + (-4.54 + 4.54i)T - 67iT^{2}
71 1+(6.79+6.79i)T+71iT2 1 + (6.79 + 6.79i)T + 71iT^{2}
73 1+(−6.04−6.04i)T+73iT2 1 + (-6.04 - 6.04i)T + 73iT^{2}
79 1−8.77T+79T2 1 - 8.77T + 79T^{2}
83 1+(8.31+8.31i)T+83iT2 1 + (8.31 + 8.31i)T + 83iT^{2}
89 1+(−9.62+9.62i)T−89iT2 1 + (-9.62 + 9.62i)T - 89iT^{2}
97 1+(−1.34+1.34i)T−97iT2 1 + (-1.34 + 1.34i)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

−10.79444371254273319893086034090, −10.09069630401630830700970479434, −9.798056300562217766067195148483, −8.649323122474220434667817545352, −7.84383224081159503928270009350, −6.18574379445520170896882035536, −5.40845413938908047879135216829, −4.65348651091447622478199383172, −2.93759930382338909971485111361, −1.93108673412243754568020864078, 0.71491655966636289917736167440, 2.41901246069312047508902490914, 3.72805124006273775740546284561, 5.32679928093137220724192564786, 6.18820325937606783469254170074, 7.41422121579503733210190282682, 7.923196833248429366137498781122, 8.523684377076861793719134593036, 9.585799990698051105209669595065, 10.81469910600148278638445135262

Graph of the ZZ-function along the critical line