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Properties

Label 2-507-507.5-c1-0-10
Degree 22
Conductor 507507
Sign −0.993+0.112i-0.993 + 0.112i
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  + (−0.367 + 2.00i)2-s + (1.27 + 1.16i)3-s + (−2.01 − 0.765i)4-s + (1.15 + 0.0695i)5-s + (−2.81 + 2.13i)6-s + (−2.33 − 1.40i)7-s + (0.166 − 0.275i)8-s + (0.271 + 2.98i)9-s + (−0.562 + 2.28i)10-s + (0.882 + 4.81i)11-s + (−1.68 − 3.33i)12-s + (3.44 − 1.06i)13-s + (3.68 − 4.15i)14-s + (1.38 + 1.43i)15-s + (−2.73 − 2.42i)16-s + (−3.64 + 0.897i)17-s + ⋯
L(s)  = 1  + (−0.259 + 1.41i)2-s + (0.738 + 0.674i)3-s + (−1.00 − 0.382i)4-s + (0.514 + 0.0311i)5-s + (−1.14 + 0.871i)6-s + (−0.881 − 0.532i)7-s + (0.0588 − 0.0974i)8-s + (0.0904 + 0.995i)9-s + (−0.177 + 0.721i)10-s + (0.266 + 1.45i)11-s + (−0.486 − 0.962i)12-s + (0.955 − 0.296i)13-s + (0.984 − 1.11i)14-s + (0.358 + 0.369i)15-s + (−0.684 − 0.606i)16-s + (−0.883 + 0.217i)17-s + ⋯

Functional equation

Λ(s)=(507s/2 ΓC(s) L(s)=((−0.993+0.112i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.112i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(507s/2 ΓC(s+1/2) L(s)=((−0.993+0.112i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 507507    =    3⋅1323 \cdot 13^{2}
Sign: −0.993+0.112i-0.993 + 0.112i
Analytic conductor: 4.048414.04841
Root analytic conductor: 2.012062.01206
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ507(5,⋅)\chi_{507} (5, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 507, ( :1/2), −0.993+0.112i)(2,\ 507,\ (\ :1/2),\ -0.993 + 0.112i)

Particular Values

L(1)L(1) ≈\approx 0.0820424−1.45010i0.0820424 - 1.45010i
L(12)L(\frac12) ≈\approx 0.0820424−1.45010i0.0820424 - 1.45010i
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+(−1.27−1.16i)T 1 + (-1.27 - 1.16i)T
13 1+(−3.44+1.06i)T 1 + (-3.44 + 1.06i)T
good2 1+(0.367−2.00i)T+(−1.87−0.709i)T2 1 + (0.367 - 2.00i)T + (-1.87 - 0.709i)T^{2}
5 1+(−1.15−0.0695i)T+(4.96+0.602i)T2 1 + (-1.15 - 0.0695i)T + (4.96 + 0.602i)T^{2}
7 1+(2.33+1.40i)T+(3.25+6.19i)T2 1 + (2.33 + 1.40i)T + (3.25 + 6.19i)T^{2}
11 1+(−0.882−4.81i)T+(−10.2+3.90i)T2 1 + (-0.882 - 4.81i)T + (-10.2 + 3.90i)T^{2}
17 1+(3.64−0.897i)T+(15.0−7.90i)T2 1 + (3.64 - 0.897i)T + (15.0 - 7.90i)T^{2}
19 1+(2.46−2.46i)T−19iT2 1 + (2.46 - 2.46i)T - 19iT^{2}
23 1+1.42T+23T2 1 + 1.42T + 23T^{2}
29 1+(−6.36+4.39i)T+(10.2−27.1i)T2 1 + (-6.36 + 4.39i)T + (10.2 - 27.1i)T^{2}
31 1+(−6.30−4.93i)T+(7.41+30.0i)T2 1 + (-6.30 - 4.93i)T + (7.41 + 30.0i)T^{2}
37 1+(−3.95+5.05i)T+(−8.85−35.9i)T2 1 + (-3.95 + 5.05i)T + (-8.85 - 35.9i)T^{2}
41 1+(−2.47−7.94i)T+(−33.7+23.2i)T2 1 + (-2.47 - 7.94i)T + (-33.7 + 23.2i)T^{2}
43 1+(−5.96−0.724i)T+(41.7+10.2i)T2 1 + (-5.96 - 0.724i)T + (41.7 + 10.2i)T^{2}
47 1+(4.03+1.81i)T+(31.1+35.1i)T2 1 + (4.03 + 1.81i)T + (31.1 + 35.1i)T^{2}
53 1+(1.80+7.32i)T+(−46.9+24.6i)T2 1 + (1.80 + 7.32i)T + (-46.9 + 24.6i)T^{2}
59 1+(−0.278+4.59i)T+(−58.5−7.11i)T2 1 + (-0.278 + 4.59i)T + (-58.5 - 7.11i)T^{2}
61 1+(0.958+0.236i)T+(54.0+28.3i)T2 1 + (0.958 + 0.236i)T + (54.0 + 28.3i)T^{2}
67 1+(−12.4−5.58i)T+(44.4+50.1i)T2 1 + (-12.4 - 5.58i)T + (44.4 + 50.1i)T^{2}
71 1+(3.79+12.1i)T+(−58.4+40.3i)T2 1 + (3.79 + 12.1i)T + (-58.4 + 40.3i)T^{2}
73 1+(−9.71+1.78i)T+(68.2−25.8i)T2 1 + (-9.71 + 1.78i)T + (68.2 - 25.8i)T^{2}
79 1+(−5.39−14.2i)T+(−59.1+52.3i)T2 1 + (-5.39 - 14.2i)T + (-59.1 + 52.3i)T^{2}
83 1+(−16.4−5.12i)T+(68.3+47.1i)T2 1 + (-16.4 - 5.12i)T + (68.3 + 47.1i)T^{2}
89 1+(3.07−3.07i)T−89iT2 1 + (3.07 - 3.07i)T - 89iT^{2}
97 1+(10.0−0.607i)T+(96.2−11.6i)T2 1 + (10.0 - 0.607i)T + (96.2 - 11.6i)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

−11.01601277605965061333168926774, −9.858993292627180859961419561448, −9.668170687830580952996654447969, −8.492861893677942348462120312762, −7.87705285667254312491472211324, −6.69584077493714586998952006512, −6.20063778061273552422931343204, −4.80925203548957468503024706972, −3.90943568153517329880312993293, −2.32474901237286295332019302614, 0.878051371575740311642408235715, 2.30223836801311582228391659328, 3.04722259831135123962551666601, 4.04314477729290411313002811022, 6.18459144450634821640469577525, 6.46848871090344095520869180716, 8.250888281580424285615276282476, 9.009173768359229350291018405050, 9.394822841089877247664810086992, 10.53031381254114089725677908187

Graph of the ZZ-function along the critical line