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Properties

Label 2-507-13.10-c1-0-8
Degree 22
Conductor 507507
Sign 0.702−0.711i0.702 - 0.711i
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  + (−2.21 + 1.28i)2-s + (−0.5 − 0.866i)3-s + (2.28 − 3.95i)4-s − 0.561i·5-s + (2.21 + 1.28i)6-s + (3.08 + 1.78i)7-s + 6.56i·8-s + (−0.499 + 0.866i)9-s + (0.719 + 1.24i)10-s + (−1.73 + i)11-s − 4.56·12-s − 9.12·14-s + (−0.486 + 0.280i)15-s + (−3.84 − 6.65i)16-s + (1.28 − 2.21i)17-s − 2.56i·18-s + ⋯
L(s)  = 1  + (−1.56 + 0.905i)2-s + (−0.288 − 0.499i)3-s + (1.14 − 1.97i)4-s − 0.251i·5-s + (0.905 + 0.522i)6-s + (1.16 + 0.673i)7-s + 2.31i·8-s + (−0.166 + 0.288i)9-s + (0.227 + 0.393i)10-s + (−0.522 + 0.301i)11-s − 1.31·12-s − 2.43·14-s + (−0.125 + 0.0724i)15-s + (−0.960 − 1.66i)16-s + (0.310 − 0.538i)17-s − 0.603i·18-s + ⋯

Functional equation

Λ(s)=(507s/2 ΓC(s) L(s)=((0.702−0.711i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(507s/2 ΓC(s+1/2) L(s)=((0.702−0.711i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 507507    =    3⋅1323 \cdot 13^{2}
Sign: 0.702−0.711i0.702 - 0.711i
Analytic conductor: 4.048414.04841
Root analytic conductor: 2.012062.01206
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ507(361,⋅)\chi_{507} (361, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 507, ( :1/2), 0.702−0.711i)(2,\ 507,\ (\ :1/2),\ 0.702 - 0.711i)

Particular Values

L(1)L(1) ≈\approx 0.612303+0.255927i0.612303 + 0.255927i
L(12)L(\frac12) ≈\approx 0.612303+0.255927i0.612303 + 0.255927i
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+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
13 1 1
good2 1+(2.21−1.28i)T+(1−1.73i)T2 1 + (2.21 - 1.28i)T + (1 - 1.73i)T^{2}
5 1+0.561iT−5T2 1 + 0.561iT - 5T^{2}
7 1+(−3.08−1.78i)T+(3.5+6.06i)T2 1 + (-3.08 - 1.78i)T + (3.5 + 6.06i)T^{2}
11 1+(1.73−i)T+(5.5−9.52i)T2 1 + (1.73 - i)T + (5.5 - 9.52i)T^{2}
17 1+(−1.28+2.21i)T+(−8.5−14.7i)T2 1 + (-1.28 + 2.21i)T + (-8.5 - 14.7i)T^{2}
19 1+(0.972+0.561i)T+(9.5+16.4i)T2 1 + (0.972 + 0.561i)T + (9.5 + 16.4i)T^{2}
23 1+(−1−1.73i)T+(−11.5+19.9i)T2 1 + (-1 - 1.73i)T + (-11.5 + 19.9i)T^{2}
29 1+(−2.84−4.92i)T+(−14.5+25.1i)T2 1 + (-2.84 - 4.92i)T + (-14.5 + 25.1i)T^{2}
31 1−1.56iT−31T2 1 - 1.56iT - 31T^{2}
37 1+(−2.97+1.71i)T+(18.5−32.0i)T2 1 + (-2.97 + 1.71i)T + (18.5 - 32.0i)T^{2}
41 1+(2.21−1.28i)T+(20.5−35.5i)T2 1 + (2.21 - 1.28i)T + (20.5 - 35.5i)T^{2}
43 1+(−0.219+0.379i)T+(−21.5−37.2i)T2 1 + (-0.219 + 0.379i)T + (-21.5 - 37.2i)T^{2}
47 1+8.24iT−47T2 1 + 8.24iT - 47T^{2}
53 1−11.6T+53T2 1 - 11.6T + 53T^{2}
59 1+(−9.63−5.56i)T+(29.5+51.0i)T2 1 + (-9.63 - 5.56i)T + (29.5 + 51.0i)T^{2}
61 1+(6.06−10.4i)T+(−30.5−52.8i)T2 1 + (6.06 - 10.4i)T + (-30.5 - 52.8i)T^{2}
67 1+(0.379−0.219i)T+(33.5−58.0i)T2 1 + (0.379 - 0.219i)T + (33.5 - 58.0i)T^{2}
71 1+(−12.1−7i)T+(35.5+61.4i)T2 1 + (-12.1 - 7i)T + (35.5 + 61.4i)T^{2}
73 1+1.87iT−73T2 1 + 1.87iT - 73T^{2}
79 1−9.56T+79T2 1 - 9.56T + 79T^{2}
83 1−9.12iT−83T2 1 - 9.12iT - 83T^{2}
89 1+(−11.3+6.56i)T+(44.5−77.0i)T2 1 + (-11.3 + 6.56i)T + (44.5 - 77.0i)T^{2}
97 1+(3.84+2.21i)T+(48.5+84.0i)T2 1 + (3.84 + 2.21i)T + (48.5 + 84.0i)T^{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

−10.81774852979569924506364770592, −9.988820380433400401719544755728, −8.815254266720033840451595568797, −8.447419428593932565464517554435, −7.48581874907980893937014822043, −6.85273883288309229155922255746, −5.59192438597954214102621499995, −5.00183407881278575081353542156, −2.31582799341852914681894039369, −1.08197688517649470736827724740, 0.903118071279828179056018451984, 2.36259700712080783233910407137, 3.67878274951395642826261104006, 4.90590418366153579172877211429, 6.49422558333374625063223148021, 7.73446155305829222216194821087, 8.194637586558895335113505887886, 9.156400396442700653367677245402, 10.19492366505295259586542301142, 10.65829342016888750166436349439

Graph of the ZZ-function along the critical line