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Properties

Label 2-253-11.10-c2-0-9
Degree 22
Conductor 253253
Sign −0.340−0.940i-0.340 - 0.940i
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  + 0.431i·2-s − 1.07·3-s + 3.81·4-s − 3.78·5-s − 0.463i·6-s + 1.98i·7-s + 3.37i·8-s − 7.84·9-s − 1.63i·10-s + (3.74 + 10.3i)11-s − 4.09·12-s + 10.8i·13-s − 0.856·14-s + 4.06·15-s + 13.8·16-s + 13.6i·17-s + ⋯
L(s)  = 1  + 0.215i·2-s − 0.358·3-s + 0.953·4-s − 0.756·5-s − 0.0772i·6-s + 0.283i·7-s + 0.421i·8-s − 0.871·9-s − 0.163i·10-s + (0.340 + 0.940i)11-s − 0.341·12-s + 0.835i·13-s − 0.0611·14-s + 0.270·15-s + 0.862·16-s + 0.803i·17-s + ⋯

Functional equation

Λ(s)=(253s/2 ΓC(s) L(s)=((−0.340−0.940i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.340 - 0.940i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(253s/2 ΓC(s+1) L(s)=((−0.340−0.940i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.340 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 253253    =    11⋅2311 \cdot 23
Sign: −0.340−0.940i-0.340 - 0.940i
Analytic conductor: 6.893756.89375
Root analytic conductor: 2.625592.62559
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ253(208,⋅)\chi_{253} (208, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 253, ( :1), −0.340−0.940i)(2,\ 253,\ (\ :1),\ -0.340 - 0.940i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.666251+0.949385i0.666251 + 0.949385i
L(12)L(\frac12) ≈\approx 0.666251+0.949385i0.666251 + 0.949385i
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+(−3.74−10.3i)T 1 + (-3.74 - 10.3i)T
23 1+4.79T 1 + 4.79T
good2 1−0.431iT−4T2 1 - 0.431iT - 4T^{2}
3 1+1.07T+9T2 1 + 1.07T + 9T^{2}
5 1+3.78T+25T2 1 + 3.78T + 25T^{2}
7 1−1.98iT−49T2 1 - 1.98iT - 49T^{2}
13 1−10.8iT−169T2 1 - 10.8iT - 169T^{2}
17 1−13.6iT−289T2 1 - 13.6iT - 289T^{2}
19 1+5.19iT−361T2 1 + 5.19iT - 361T^{2}
29 1−55.1iT−841T2 1 - 55.1iT - 841T^{2}
31 1+0.0141T+961T2 1 + 0.0141T + 961T^{2}
37 1+3.33T+1.36e3T2 1 + 3.33T + 1.36e3T^{2}
41 1+12.2iT−1.68e3T2 1 + 12.2iT - 1.68e3T^{2}
43 1−0.275iT−1.84e3T2 1 - 0.275iT - 1.84e3T^{2}
47 1+65.1T+2.20e3T2 1 + 65.1T + 2.20e3T^{2}
53 1−6.34T+2.80e3T2 1 - 6.34T + 2.80e3T^{2}
59 1−2.24T+3.48e3T2 1 - 2.24T + 3.48e3T^{2}
61 1+81.8iT−3.72e3T2 1 + 81.8iT - 3.72e3T^{2}
67 1−36.3T+4.48e3T2 1 - 36.3T + 4.48e3T^{2}
71 1−54.9T+5.04e3T2 1 - 54.9T + 5.04e3T^{2}
73 1+70.4iT−5.32e3T2 1 + 70.4iT - 5.32e3T^{2}
79 1+105.iT−6.24e3T2 1 + 105. iT - 6.24e3T^{2}
83 1−39.4iT−6.88e3T2 1 - 39.4iT - 6.88e3T^{2}
89 1−73.9T+7.92e3T2 1 - 73.9T + 7.92e3T^{2}
97 1−47.9T+9.40e3T2 1 - 47.9T + 9.40e3T^{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

−11.93879173483847517391524821647, −11.33913218256131308583492960709, −10.46853006107978624575186921465, −9.084480211818938385398500355323, −8.060885144961316918725155425198, −7.03188010513066811158141748947, −6.22840229144556150315435301613, −4.98745637307853400873892210617, −3.50596762190467335461259854644, −1.94492227284247683665236170164, 0.60393639772095317230579168136, 2.72963497503795060153477616871, 3.81380563216228147197269599477, 5.54075036021000908090973052271, 6.40101980880933382993347093661, 7.63619474098815650590354990283, 8.353241740430430316750408835551, 9.851087464131271696175332822004, 10.88218563008112405359940047994, 11.59152808767632855185407153807

Graph of the ZZ-function along the critical line