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Properties

Label 2-507-39.5-c1-0-19
Degree 22
Conductor 507507
Sign −0.957−0.289i-0.957 - 0.289i
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  + (−1.69 − 1.69i)2-s − 1.73·3-s + 3.73i·4-s + (−1.23 − 1.23i)5-s + (2.93 + 2.93i)6-s + (2.93 − 2.93i)8-s + 2.99·9-s + 4.19i·10-s + (4.62 − 4.62i)11-s − 6.46i·12-s + (2.14 + 2.14i)15-s − 2.46·16-s + (−5.07 − 5.07i)18-s + (4.62 − 4.62i)20-s − 15.6·22-s + ⋯
L(s)  = 1  + (−1.19 − 1.19i)2-s − 1.00·3-s + 1.86i·4-s + (−0.554 − 0.554i)5-s + (1.19 + 1.19i)6-s + (1.03 − 1.03i)8-s + 0.999·9-s + 1.32i·10-s + (1.39 − 1.39i)11-s − 1.86i·12-s + (0.554 + 0.554i)15-s − 0.616·16-s + (−1.19 − 1.19i)18-s + (1.03 − 1.03i)20-s − 3.33·22-s + ⋯

Functional equation

Λ(s)=(507s/2 ΓC(s) L(s)=((−0.957−0.289i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(507s/2 ΓC(s+1/2) L(s)=((−0.957−0.289i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 507507    =    3⋅1323 \cdot 13^{2}
Sign: −0.957−0.289i-0.957 - 0.289i
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), −0.957−0.289i)(2,\ 507,\ (\ :1/2),\ -0.957 - 0.289i)

Particular Values

L(1)L(1) ≈\approx 0.0524409+0.354166i0.0524409 + 0.354166i
L(12)L(\frac12) ≈\approx 0.0524409+0.354166i0.0524409 + 0.354166i
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+1.73T 1 + 1.73T
13 1 1
good2 1+(1.69+1.69i)T+2iT2 1 + (1.69 + 1.69i)T + 2iT^{2}
5 1+(1.23+1.23i)T+5iT2 1 + (1.23 + 1.23i)T + 5iT^{2}
7 1+7iT2 1 + 7iT^{2}
11 1+(−4.62+4.62i)T−11iT2 1 + (-4.62 + 4.62i)T - 11iT^{2}
17 1+17T2 1 + 17T^{2}
19 1−19iT2 1 - 19iT^{2}
23 1+23T2 1 + 23T^{2}
29 1−29T2 1 - 29T^{2}
31 1−31iT2 1 - 31iT^{2}
37 1+37iT2 1 + 37iT^{2}
41 1+(5.53+5.53i)T+41iT2 1 + (5.53 + 5.53i)T + 41iT^{2}
43 1−4iT−43T2 1 - 4iT - 43T^{2}
47 1+(7.10−7.10i)T−47iT2 1 + (7.10 - 7.10i)T - 47iT^{2}
53 1−53T2 1 - 53T^{2}
59 1+(0.332−0.332i)T−59iT2 1 + (0.332 - 0.332i)T - 59iT^{2}
61 1−13.8T+61T2 1 - 13.8T + 61T^{2}
67 1−67iT2 1 - 67iT^{2}
71 1+(11.3+11.3i)T+71iT2 1 + (11.3 + 11.3i)T + 71iT^{2}
73 1+73iT2 1 + 73iT^{2}
79 1+10.3T+79T2 1 + 10.3T + 79T^{2}
83 1+(8.91+8.91i)T+83iT2 1 + (8.91 + 8.91i)T + 83iT^{2}
89 1+(3.05−3.05i)T−89iT2 1 + (3.05 - 3.05i)T - 89iT^{2}
97 1−97iT2 1 - 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

−10.55224015262241203045291256931, −9.670151954331657137084752914666, −8.795127409385298445395348240656, −8.153914594461172768075226261279, −6.90096393378889863103559155319, −5.80308974991411047295512309273, −4.33789755386919939016430319827, −3.37124666151352713988568052088, −1.47998144922730687466026414876, −0.41867206179546221929824922636, 1.43933373722273104205715592922, 3.98152206935158859395202247106, 5.16508319906434781603128638315, 6.34171707869110473580380117646, 6.96235816735254795831297167934, 7.45085446184231745133837866199, 8.651598694586480859741007533325, 9.695153232119295266080528387262, 10.14360053890275136933341888512, 11.28226484144919312067932750781

Graph of the ZZ-function along the critical line