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Properties

Label 2-507-169.16-c1-0-8
Degree 22
Conductor 507507
Sign −0.143−0.989i-0.143 - 0.989i
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  + (−0.164 − 0.0132i)2-s + (0.845 + 0.534i)3-s + (−1.94 − 0.316i)4-s + (2.22 + 1.96i)5-s + (−0.131 − 0.0989i)6-s + (−0.158 + 0.164i)7-s + (0.635 + 0.156i)8-s + (0.428 + 0.903i)9-s + (−0.338 − 0.352i)10-s + (−2.22 + 4.68i)11-s + (−1.47 − 1.30i)12-s + (−3.60 − 0.160i)13-s + (0.0281 − 0.0249i)14-s + (0.826 + 2.85i)15-s + (3.64 + 1.21i)16-s + (−1.50 + 1.56i)17-s + ⋯
L(s)  = 1  + (−0.116 − 0.00936i)2-s + (0.487 + 0.308i)3-s + (−0.973 − 0.158i)4-s + (0.994 + 0.880i)5-s + (−0.0537 − 0.0403i)6-s + (−0.0597 + 0.0621i)7-s + (0.224 + 0.0553i)8-s + (0.142 + 0.301i)9-s + (−0.107 − 0.111i)10-s + (−0.670 + 1.41i)11-s + (−0.426 − 0.377i)12-s + (−0.999 − 0.0443i)13-s + (0.00751 − 0.00665i)14-s + (0.213 + 0.736i)15-s + (0.910 + 0.303i)16-s + (−0.364 + 0.379i)17-s + ⋯

Functional equation

Λ(s)=(507s/2 ΓC(s) L(s)=((−0.143−0.989i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.143 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(507s/2 ΓC(s+1/2) L(s)=((−0.143−0.989i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.143 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 507507    =    3⋅1323 \cdot 13^{2}
Sign: −0.143−0.989i-0.143 - 0.989i
Analytic conductor: 4.048414.04841
Root analytic conductor: 2.012062.01206
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ507(16,⋅)\chi_{507} (16, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 507, ( :1/2), −0.143−0.989i)(2,\ 507,\ (\ :1/2),\ -0.143 - 0.989i)

Particular Values

L(1)L(1) ≈\approx 0.827246+0.955433i0.827246 + 0.955433i
L(12)L(\frac12) ≈\approx 0.827246+0.955433i0.827246 + 0.955433i
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+(−0.845−0.534i)T 1 + (-0.845 - 0.534i)T
13 1+(3.60+0.160i)T 1 + (3.60 + 0.160i)T
good2 1+(0.164+0.0132i)T+(1.97+0.320i)T2 1 + (0.164 + 0.0132i)T + (1.97 + 0.320i)T^{2}
5 1+(−2.22−1.96i)T+(0.602+4.96i)T2 1 + (-2.22 - 1.96i)T + (0.602 + 4.96i)T^{2}
7 1+(0.158−0.164i)T+(−0.281−6.99i)T2 1 + (0.158 - 0.164i)T + (-0.281 - 6.99i)T^{2}
11 1+(2.22−4.68i)T+(−6.95−8.52i)T2 1 + (2.22 - 4.68i)T + (-6.95 - 8.52i)T^{2}
17 1+(1.50−1.56i)T+(−0.684−16.9i)T2 1 + (1.50 - 1.56i)T + (-0.684 - 16.9i)T^{2}
19 1+(−2.38+4.13i)T+(−9.5−16.4i)T2 1 + (-2.38 + 4.13i)T + (-9.5 - 16.4i)T^{2}
23 1+(−1.04−1.81i)T+(−11.5+19.9i)T2 1 + (-1.04 - 1.81i)T + (-11.5 + 19.9i)T^{2}
29 1+(−5.67−0.458i)T+(28.6+4.65i)T2 1 + (-5.67 - 0.458i)T + (28.6 + 4.65i)T^{2}
31 1+(0.846−6.97i)T+(−30.0−7.41i)T2 1 + (0.846 - 6.97i)T + (-30.0 - 7.41i)T^{2}
37 1+(10.5−4.48i)T+(25.6−26.6i)T2 1 + (10.5 - 4.48i)T + (25.6 - 26.6i)T^{2}
41 1+(0.450+0.284i)T+(17.5+37.0i)T2 1 + (0.450 + 0.284i)T + (17.5 + 37.0i)T^{2}
43 1+(−3.19−1.35i)T+(29.7+31.0i)T2 1 + (-3.19 - 1.35i)T + (29.7 + 31.0i)T^{2}
47 1+(−0.481+1.26i)T+(−35.1−31.1i)T2 1 + (-0.481 + 1.26i)T + (-35.1 - 31.1i)T^{2}
53 1+(−7.73−1.90i)T+(46.9+24.6i)T2 1 + (-7.73 - 1.90i)T + (46.9 + 24.6i)T^{2}
59 1+(−8.17+2.72i)T+(47.1−35.4i)T2 1 + (-8.17 + 2.72i)T + (47.1 - 35.4i)T^{2}
61 1+(−0.807+2.78i)T+(−51.5−32.6i)T2 1 + (-0.807 + 2.78i)T + (-51.5 - 32.6i)T^{2}
67 1+(−14.6+2.37i)T+(63.5−21.2i)T2 1 + (-14.6 + 2.37i)T + (63.5 - 21.2i)T^{2}
71 1+(−0.600+14.8i)T+(−70.7−5.71i)T2 1 + (-0.600 + 14.8i)T + (-70.7 - 5.71i)T^{2}
73 1+(4.24+6.14i)T+(−25.8+68.2i)T2 1 + (4.24 + 6.14i)T + (-25.8 + 68.2i)T^{2}
79 1+(−3.43+9.06i)T+(−59.1−52.3i)T2 1 + (-3.43 + 9.06i)T + (-59.1 - 52.3i)T^{2}
83 1+(13.0+6.86i)T+(47.1+68.3i)T2 1 + (13.0 + 6.86i)T + (47.1 + 68.3i)T^{2}
89 1+(−3.64−6.31i)T+(−44.5+77.0i)T2 1 + (-3.64 - 6.31i)T + (-44.5 + 77.0i)T^{2}
97 1+(−2.65−13.0i)T+(−89.2+38.0i)T2 1 + (-2.65 - 13.0i)T + (-89.2 + 38.0i)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

−10.59436451643592922957550424493, −10.16201608746184419608202456187, −9.535606279861345973466933647521, −8.736129990699720549325032591836, −7.51301294705584200241782937015, −6.71952435022455118375619374419, −5.24032624395498879105324194974, −4.68425820872793806605355807094, −3.12183688976186189509702274200, −2.04266712301366389545408345623, 0.76385840225329433135219204053, 2.45591847638019696347119629287, 3.83961002827845736508677708043, 5.16733375625855659332803919000, 5.69518442523093332759195052691, 7.20628425784043717436542918731, 8.371480024947177746030809503552, 8.721071673501670854347724665281, 9.718214459639571750079933177260, 10.23563400762068755182749286668

Graph of the ZZ-function along the critical line