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Properties

Label 2-1352-13.4-c1-0-22
Degree 22
Conductor 13521352
Sign 0.890−0.454i0.890 - 0.454i
Analytic cond. 10.795710.7957
Root an. cond. 3.285693.28569
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  + (−0.5 + 0.866i)3-s + i·5-s + (4.33 − 2.5i)7-s + (1 + 1.73i)9-s + (1.73 + i)11-s + (−0.866 − 0.5i)15-s + (−1.5 − 2.59i)17-s + (1.73 − i)19-s + 5i·21-s + (2 − 3.46i)23-s + 4·25-s − 5·27-s + (3 − 5.19i)29-s + 4i·31-s + (−1.73 + 0.999i)33-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + 0.447i·5-s + (1.63 − 0.944i)7-s + (0.333 + 0.577i)9-s + (0.522 + 0.301i)11-s + (−0.223 − 0.129i)15-s + (−0.363 − 0.630i)17-s + (0.397 − 0.229i)19-s + 1.09i·21-s + (0.417 − 0.722i)23-s + 0.800·25-s − 0.962·27-s + (0.557 − 0.964i)29-s + 0.718i·31-s + (−0.301 + 0.174i)33-s + ⋯

Functional equation

Λ(s)=(1352s/2 ΓC(s) L(s)=((0.890−0.454i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.454i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1352s/2 ΓC(s+1/2) L(s)=((0.890−0.454i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.890 - 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13521352    =    23⋅1322^{3} \cdot 13^{2}
Sign: 0.890−0.454i0.890 - 0.454i
Analytic conductor: 10.795710.7957
Root analytic conductor: 3.285693.28569
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1352(1161,⋅)\chi_{1352} (1161, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1352, ( :1/2), 0.890−0.454i)(2,\ 1352,\ (\ :1/2),\ 0.890 - 0.454i)

Particular Values

L(1)L(1) ≈\approx 1.9870370381.987037038
L(12)L(\frac12) ≈\approx 1.9870370381.987037038
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)
bad2 1 1
13 1 1
good3 1+(0.5−0.866i)T+(−1.5−2.59i)T2 1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2}
5 1−iT−5T2 1 - iT - 5T^{2}
7 1+(−4.33+2.5i)T+(3.5−6.06i)T2 1 + (-4.33 + 2.5i)T + (3.5 - 6.06i)T^{2}
11 1+(−1.73−i)T+(5.5+9.52i)T2 1 + (-1.73 - i)T + (5.5 + 9.52i)T^{2}
17 1+(1.5+2.59i)T+(−8.5+14.7i)T2 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2}
19 1+(−1.73+i)T+(9.5−16.4i)T2 1 + (-1.73 + i)T + (9.5 - 16.4i)T^{2}
23 1+(−2+3.46i)T+(−11.5−19.9i)T2 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2}
29 1+(−3+5.19i)T+(−14.5−25.1i)T2 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2}
31 1−4iT−31T2 1 - 4iT - 31T^{2}
37 1+(9.52+5.5i)T+(18.5+32.0i)T2 1 + (9.52 + 5.5i)T + (18.5 + 32.0i)T^{2}
41 1+(−6.92−4i)T+(20.5+35.5i)T2 1 + (-6.92 - 4i)T + (20.5 + 35.5i)T^{2}
43 1+(0.5+0.866i)T+(−21.5+37.2i)T2 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2}
47 1−9iT−47T2 1 - 9iT - 47T^{2}
53 1+12T+53T2 1 + 12T + 53T^{2}
59 1+(−5.19+3i)T+(29.5−51.0i)T2 1 + (-5.19 + 3i)T + (29.5 - 51.0i)T^{2}
61 1+(−30.5+52.8i)T2 1 + (-30.5 + 52.8i)T^{2}
67 1+(−5.19−3i)T+(33.5+58.0i)T2 1 + (-5.19 - 3i)T + (33.5 + 58.0i)T^{2}
71 1+(6.06−3.5i)T+(35.5−61.4i)T2 1 + (6.06 - 3.5i)T + (35.5 - 61.4i)T^{2}
73 1+2iT−73T2 1 + 2iT - 73T^{2}
79 1−12T+79T2 1 - 12T + 79T^{2}
83 1−16iT−83T2 1 - 16iT - 83T^{2}
89 1+(−8.66−5i)T+(44.5+77.0i)T2 1 + (-8.66 - 5i)T + (44.5 + 77.0i)T^{2}
97 1+(−8.66+5i)T+(48.5−84.0i)T2 1 + (-8.66 + 5i)T + (48.5 - 84.0i)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

−9.828396482441445330519962602808, −8.869595167360800162528981679554, −7.904633912967233763830519555072, −7.30807965185695685931448855228, −6.52494852169395697927909636057, −5.04189248480755930685929310993, −4.74713927219850947945654995439, −3.85725031879058091484370889271, −2.38939388308977133714152700041, −1.16326587446068106245435608355, 1.17454344268463747384849374258, 1.90922393461162340517652351645, 3.45704971422960311177900691371, 4.65142102538465163895346004213, 5.33756101457568575114146852483, 6.18719537761229473246899212685, 7.11584689957780865487201945311, 8.013229817688853375932979838742, 8.764269641057153908610105137540, 9.217208960672570198340339327360

Graph of the ZZ-function along the critical line