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Properties

Label 2-253-11.2-c2-0-12
Degree 22
Conductor 253253
Sign −0.0488−0.998i-0.0488 - 0.998i
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.675 + 0.219i)2-s + (1.44 − 1.05i)3-s + (−2.82 − 2.05i)4-s + (1.51 + 4.64i)5-s + (1.20 − 0.392i)6-s + (−5.17 + 7.12i)7-s + (−3.13 − 4.30i)8-s + (−1.79 + 5.51i)9-s + 3.47i·10-s + (−2.65 + 10.6i)11-s − 6.25·12-s + (7.38 + 2.40i)13-s + (−5.05 + 3.67i)14-s + (7.07 + 5.13i)15-s + (3.15 + 9.69i)16-s + (7.23 − 2.35i)17-s + ⋯
L(s)  = 1  + (0.337 + 0.109i)2-s + (0.482 − 0.350i)3-s + (−0.706 − 0.513i)4-s + (0.302 + 0.929i)5-s + (0.201 − 0.0654i)6-s + (−0.739 + 1.01i)7-s + (−0.391 − 0.538i)8-s + (−0.199 + 0.613i)9-s + 0.347i·10-s + (−0.240 + 0.970i)11-s − 0.520·12-s + (0.568 + 0.184i)13-s + (−0.361 + 0.262i)14-s + (0.471 + 0.342i)15-s + (0.196 + 0.606i)16-s + (0.425 − 0.138i)17-s + ⋯

Functional equation

Λ(s)=(253s/2 ΓC(s) L(s)=((−0.0488−0.998i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0488 - 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(253s/2 ΓC(s+1) L(s)=((−0.0488−0.998i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0488 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 253253    =    11⋅2311 \cdot 23
Sign: −0.0488−0.998i-0.0488 - 0.998i
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), −0.0488−0.998i)(2,\ 253,\ (\ :1),\ -0.0488 - 0.998i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.990982+1.04061i0.990982 + 1.04061i
L(12)L(\frac12) ≈\approx 0.990982+1.04061i0.990982 + 1.04061i
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+(2.65−10.6i)T 1 + (2.65 - 10.6i)T
23 1+4.79T 1 + 4.79T
good2 1+(−0.675−0.219i)T+(3.23+2.35i)T2 1 + (-0.675 - 0.219i)T + (3.23 + 2.35i)T^{2}
3 1+(−1.44+1.05i)T+(2.78−8.55i)T2 1 + (-1.44 + 1.05i)T + (2.78 - 8.55i)T^{2}
5 1+(−1.51−4.64i)T+(−20.2+14.6i)T2 1 + (-1.51 - 4.64i)T + (-20.2 + 14.6i)T^{2}
7 1+(5.17−7.12i)T+(−15.1−46.6i)T2 1 + (5.17 - 7.12i)T + (-15.1 - 46.6i)T^{2}
13 1+(−7.38−2.40i)T+(136.+99.3i)T2 1 + (-7.38 - 2.40i)T + (136. + 99.3i)T^{2}
17 1+(−7.23+2.35i)T+(233.−169.i)T2 1 + (-7.23 + 2.35i)T + (233. - 169. i)T^{2}
19 1+(2.74+3.78i)T+(−111.+343.i)T2 1 + (2.74 + 3.78i)T + (-111. + 343. i)T^{2}
29 1+(3.01−4.15i)T+(−259.−799.i)T2 1 + (3.01 - 4.15i)T + (-259. - 799. i)T^{2}
31 1+(−1.25+3.86i)T+(−777.−564.i)T2 1 + (-1.25 + 3.86i)T + (-777. - 564. i)T^{2}
37 1+(12.0+8.76i)T+(423.+1.30e3i)T2 1 + (12.0 + 8.76i)T + (423. + 1.30e3i)T^{2}
41 1+(10.7+14.7i)T+(−519.+1.59e3i)T2 1 + (10.7 + 14.7i)T + (-519. + 1.59e3i)T^{2}
43 1−10.7iT−1.84e3T2 1 - 10.7iT - 1.84e3T^{2}
47 1+(54.0−39.2i)T+(682.−2.10e3i)T2 1 + (54.0 - 39.2i)T + (682. - 2.10e3i)T^{2}
53 1+(0.202−0.623i)T+(−2.27e3−1.65e3i)T2 1 + (0.202 - 0.623i)T + (-2.27e3 - 1.65e3i)T^{2}
59 1+(62.6+45.5i)T+(1.07e3+3.31e3i)T2 1 + (62.6 + 45.5i)T + (1.07e3 + 3.31e3i)T^{2}
61 1+(−77.4+25.1i)T+(3.01e3−2.18e3i)T2 1 + (-77.4 + 25.1i)T + (3.01e3 - 2.18e3i)T^{2}
67 1−41.4T+4.48e3T2 1 - 41.4T + 4.48e3T^{2}
71 1+(−23.6−72.8i)T+(−4.07e3+2.96e3i)T2 1 + (-23.6 - 72.8i)T + (-4.07e3 + 2.96e3i)T^{2}
73 1+(3.57−4.91i)T+(−1.64e3−5.06e3i)T2 1 + (3.57 - 4.91i)T + (-1.64e3 - 5.06e3i)T^{2}
79 1+(−88.9−28.9i)T+(5.04e3+3.66e3i)T2 1 + (-88.9 - 28.9i)T + (5.04e3 + 3.66e3i)T^{2}
83 1+(−125.+40.8i)T+(5.57e3−4.04e3i)T2 1 + (-125. + 40.8i)T + (5.57e3 - 4.04e3i)T^{2}
89 1−57.8T+7.92e3T2 1 - 57.8T + 7.92e3T^{2}
97 1+(−22.7+70.1i)T+(−7.61e3−5.53e3i)T2 1 + (-22.7 + 70.1i)T + (-7.61e3 - 5.53e3i)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

−12.40938490685175529860275938169, −10.99105120383017387188498784223, −10.00101857910464453079054377055, −9.265031321560848723585432851863, −8.222671016214993784066755271383, −6.89472219461797488333695240952, −6.00235559440981905045977742583, −4.94028907211232758773291241294, −3.30226194190774658881672090694, −2.13201398429845879038364548696, 0.64702534168029926299645752070, 3.27732921779551814231131949353, 3.86619173309545822987493891690, 5.16230634223610519060447231930, 6.34894647646015010485980645009, 7.981148689181903170247083593991, 8.726389262451708997130347608023, 9.478093761782610405203780863500, 10.43559662901108973110726146962, 11.80363062811481378076820668349

Graph of the ZZ-function along the critical line