↑

Properties

Label 2-253-11.2-c2-0-13
Degree 22
Conductor 253253
Sign 0.508−0.860i0.508 - 0.860i
Analytic cond. 6.893756.89375
Root an. cond. 2.625592.62559
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.81 − 0.913i)2-s + (−3.21 + 2.33i)3-s + (3.83 + 2.78i)4-s + (1.38 + 4.25i)5-s + (11.1 − 3.62i)6-s + (3.16 − 4.35i)7-s + (−1.29 − 1.78i)8-s + (2.09 − 6.43i)9-s − 13.2i·10-s + (8.08 − 7.45i)11-s − 18.8·12-s + (−16.0 − 5.21i)13-s + (−12.8 + 9.36i)14-s + (−14.3 − 10.4i)15-s + (−3.85 − 11.8i)16-s + (15.0 − 4.89i)17-s + ⋯
L(s)  = 1  + (−1.40 − 0.456i)2-s + (−1.07 + 0.777i)3-s + (0.959 + 0.697i)4-s + (0.276 + 0.850i)5-s + (1.86 − 0.604i)6-s + (0.452 − 0.622i)7-s + (−0.162 − 0.223i)8-s + (0.232 − 0.714i)9-s − 1.32i·10-s + (0.735 − 0.677i)11-s − 1.57·12-s + (−1.23 − 0.400i)13-s + (−0.920 + 0.668i)14-s + (−0.957 − 0.695i)15-s + (−0.240 − 0.740i)16-s + (0.886 − 0.287i)17-s + ⋯

Functional equation

Λ(s)=(253s/2 ΓC(s) L(s)=((0.508−0.860i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.508 - 0.860i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(253s/2 ΓC(s+1) L(s)=((0.508−0.860i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.508 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 253253    =    11⋅2311 \cdot 23
Sign: 0.508−0.860i0.508 - 0.860i
Analytic conductor: 6.893756.89375
Root analytic conductor: 2.625592.62559
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ253(24,⋅)\chi_{253} (24, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 253, ( :1), 0.508−0.860i)(2,\ 253,\ (\ :1),\ 0.508 - 0.860i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.467343+0.266636i0.467343 + 0.266636i
L(12)L(\frac12) ≈\approx 0.467343+0.266636i0.467343 + 0.266636i
L(2)L(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)
bad11 1+(−8.08+7.45i)T 1 + (-8.08 + 7.45i)T
23 1−4.79T 1 - 4.79T
good2 1+(2.81+0.913i)T+(3.23+2.35i)T2 1 + (2.81 + 0.913i)T + (3.23 + 2.35i)T^{2}
3 1+(3.21−2.33i)T+(2.78−8.55i)T2 1 + (3.21 - 2.33i)T + (2.78 - 8.55i)T^{2}
5 1+(−1.38−4.25i)T+(−20.2+14.6i)T2 1 + (-1.38 - 4.25i)T + (-20.2 + 14.6i)T^{2}
7 1+(−3.16+4.35i)T+(−15.1−46.6i)T2 1 + (-3.16 + 4.35i)T + (-15.1 - 46.6i)T^{2}
13 1+(16.0+5.21i)T+(136.+99.3i)T2 1 + (16.0 + 5.21i)T + (136. + 99.3i)T^{2}
17 1+(−15.0+4.89i)T+(233.−169.i)T2 1 + (-15.0 + 4.89i)T + (233. - 169. i)T^{2}
19 1+(−18.5−25.5i)T+(−111.+343.i)T2 1 + (-18.5 - 25.5i)T + (-111. + 343. i)T^{2}
29 1+(−17.8+24.6i)T+(−259.−799.i)T2 1 + (-17.8 + 24.6i)T + (-259. - 799. i)T^{2}
31 1+(13.7−42.2i)T+(−777.−564.i)T2 1 + (13.7 - 42.2i)T + (-777. - 564. i)T^{2}
37 1+(4.03+2.93i)T+(423.+1.30e3i)T2 1 + (4.03 + 2.93i)T + (423. + 1.30e3i)T^{2}
41 1+(−41.3−56.9i)T+(−519.+1.59e3i)T2 1 + (-41.3 - 56.9i)T + (-519. + 1.59e3i)T^{2}
43 1−54.5iT−1.84e3T2 1 - 54.5iT - 1.84e3T^{2}
47 1+(9.04−6.57i)T+(682.−2.10e3i)T2 1 + (9.04 - 6.57i)T + (682. - 2.10e3i)T^{2}
53 1+(−6.48+19.9i)T+(−2.27e3−1.65e3i)T2 1 + (-6.48 + 19.9i)T + (-2.27e3 - 1.65e3i)T^{2}
59 1+(59.2+43.0i)T+(1.07e3+3.31e3i)T2 1 + (59.2 + 43.0i)T + (1.07e3 + 3.31e3i)T^{2}
61 1+(−30.6+9.96i)T+(3.01e3−2.18e3i)T2 1 + (-30.6 + 9.96i)T + (3.01e3 - 2.18e3i)T^{2}
67 1+66.0T+4.48e3T2 1 + 66.0T + 4.48e3T^{2}
71 1+(−22.6−69.6i)T+(−4.07e3+2.96e3i)T2 1 + (-22.6 - 69.6i)T + (-4.07e3 + 2.96e3i)T^{2}
73 1+(77.8−107.i)T+(−1.64e3−5.06e3i)T2 1 + (77.8 - 107. i)T + (-1.64e3 - 5.06e3i)T^{2}
79 1+(15.1+4.90i)T+(5.04e3+3.66e3i)T2 1 + (15.1 + 4.90i)T + (5.04e3 + 3.66e3i)T^{2}
83 1+(−6.22+2.02i)T+(5.57e3−4.04e3i)T2 1 + (-6.22 + 2.02i)T + (5.57e3 - 4.04e3i)T^{2}
89 1−52.0T+7.92e3T2 1 - 52.0T + 7.92e3T^{2}
97 1+(4.28−13.1i)T+(−7.61e3−5.53e3i)T2 1 + (4.28 - 13.1i)T + (-7.61e3 - 5.53e3i)T^{2}
show more
show less
   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

−11.46021629782829226934493668243, −10.81434696319182283295773789105, −10.01926377377896841967085093410, −9.695597444480865373741886799668, −8.123640782925085779124594521248, −7.26279753555382033255388512412, −5.94755902397906685090613647496, −4.75862780817747009544192560400, −3.05403378855627756868303237545, −1.06795162476902262614131638490, 0.67061805038389503287564807339, 1.81434037093056131838940782472, 4.82069852907004932613693612158, 5.76087191331044055662867717081, 7.03409717000178973070660370028, 7.50207359532486480994057459385, 8.994814772348613689354405105041, 9.325347569311016096878203730275, 10.57007814470436938282021650397, 11.80576841425591081172100608739

Graph of the ZZ-function along the critical line