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Properties

Label 2-296-296.101-c1-0-21
Degree 22
Conductor 296296
Sign −0.901+0.431i-0.901 + 0.431i
Analytic cond. 2.363572.36357
Root an. cond. 1.537391.53739
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  + (−1.17 − 0.781i)2-s + (−0.593 + 0.342i)3-s + (0.778 + 1.84i)4-s + (−1.02 − 1.77i)5-s + (0.967 + 0.0599i)6-s + (0.355 + 0.615i)7-s + (0.521 − 2.77i)8-s + (−1.26 + 2.19i)9-s + (−0.179 + 2.89i)10-s − 5.64i·11-s + (−1.09 − 0.826i)12-s + (0.935 + 1.62i)13-s + (0.0621 − 1.00i)14-s + (1.21 + 0.703i)15-s + (−2.78 + 2.86i)16-s + (−5.77 − 3.33i)17-s + ⋯
L(s)  = 1  + (−0.833 − 0.552i)2-s + (−0.342 + 0.197i)3-s + (0.389 + 0.921i)4-s + (−0.458 − 0.794i)5-s + (0.395 + 0.0244i)6-s + (0.134 + 0.232i)7-s + (0.184 − 0.982i)8-s + (−0.421 + 0.730i)9-s + (−0.0567 + 0.915i)10-s − 1.70i·11-s + (−0.315 − 0.238i)12-s + (0.259 + 0.449i)13-s + (0.0166 − 0.268i)14-s + (0.314 + 0.181i)15-s + (−0.696 + 0.717i)16-s + (−1.39 − 0.807i)17-s + ⋯

Functional equation

Λ(s)=(296s/2 ΓC(s) L(s)=((−0.901+0.431i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.901 + 0.431i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(296s/2 ΓC(s+1/2) L(s)=((−0.901+0.431i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.901 + 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 296296    =    23⋅372^{3} \cdot 37
Sign: −0.901+0.431i-0.901 + 0.431i
Analytic conductor: 2.363572.36357
Root analytic conductor: 1.537391.53739
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ296(101,⋅)\chi_{296} (101, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 296, ( :1/2), −0.901+0.431i)(2,\ 296,\ (\ :1/2),\ -0.901 + 0.431i)

Particular Values

L(1)L(1) ≈\approx 0.0816881−0.359674i0.0816881 - 0.359674i
L(12)L(\frac12) ≈\approx 0.0816881−0.359674i0.0816881 - 0.359674i
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+(1.17+0.781i)T 1 + (1.17 + 0.781i)T
37 1+(5.59−2.38i)T 1 + (5.59 - 2.38i)T
good3 1+(0.593−0.342i)T+(1.5−2.59i)T2 1 + (0.593 - 0.342i)T + (1.5 - 2.59i)T^{2}
5 1+(1.02+1.77i)T+(−2.5+4.33i)T2 1 + (1.02 + 1.77i)T + (-2.5 + 4.33i)T^{2}
7 1+(−0.355−0.615i)T+(−3.5+6.06i)T2 1 + (-0.355 - 0.615i)T + (-3.5 + 6.06i)T^{2}
11 1+5.64iT−11T2 1 + 5.64iT - 11T^{2}
13 1+(−0.935−1.62i)T+(−6.5+11.2i)T2 1 + (-0.935 - 1.62i)T + (-6.5 + 11.2i)T^{2}
17 1+(5.77+3.33i)T+(8.5+14.7i)T2 1 + (5.77 + 3.33i)T + (8.5 + 14.7i)T^{2}
19 1+(2.49+4.32i)T+(−9.5+16.4i)T2 1 + (2.49 + 4.32i)T + (-9.5 + 16.4i)T^{2}
23 1−2.13iT−23T2 1 - 2.13iT - 23T^{2}
29 1+9.00T+29T2 1 + 9.00T + 29T^{2}
31 1+6.96iT−31T2 1 + 6.96iT - 31T^{2}
41 1+(−3.42−5.93i)T+(−20.5+35.5i)T2 1 + (-3.42 - 5.93i)T + (-20.5 + 35.5i)T^{2}
43 1+6.29T+43T2 1 + 6.29T + 43T^{2}
47 1+0.475T+47T2 1 + 0.475T + 47T^{2}
53 1+(−10.1−5.85i)T+(26.5+45.8i)T2 1 + (-10.1 - 5.85i)T + (26.5 + 45.8i)T^{2}
59 1+(−5.17+8.96i)T+(−29.5−51.0i)T2 1 + (-5.17 + 8.96i)T + (-29.5 - 51.0i)T^{2}
61 1+(−3.75−6.50i)T+(−30.5+52.8i)T2 1 + (-3.75 - 6.50i)T + (-30.5 + 52.8i)T^{2}
67 1+(−2.00+1.15i)T+(33.5−58.0i)T2 1 + (-2.00 + 1.15i)T + (33.5 - 58.0i)T^{2}
71 1+(1.37+2.38i)T+(−35.5+61.4i)T2 1 + (1.37 + 2.38i)T + (-35.5 + 61.4i)T^{2}
73 1−4.17T+73T2 1 - 4.17T + 73T^{2}
79 1+(−13.8+8.02i)T+(39.5−68.4i)T2 1 + (-13.8 + 8.02i)T + (39.5 - 68.4i)T^{2}
83 1+(−6.57−3.79i)T+(41.5+71.8i)T2 1 + (-6.57 - 3.79i)T + (41.5 + 71.8i)T^{2}
89 1+(2.40+1.38i)T+(44.5+77.0i)T2 1 + (2.40 + 1.38i)T + (44.5 + 77.0i)T^{2}
97 1−3.70iT−97T2 1 - 3.70iT - 97T^{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

−11.38707701912386542128876344415, −10.72132058205843374591549199384, −9.204691677608356398158724185792, −8.706202703916002446447854376547, −7.911561870457990868318902666266, −6.53883659157406213352341778832, −5.17424770449137459232020341007, −3.94315275637081809090197679650, −2.38997653918307609200328063759, −0.35442040400165695902999697219, 1.97653475703771875816134888234, 3.92749557819741820079083849492, 5.48462861497829044418240152805, 6.72319324537084471817397618947, 7.10603595800295529332434858774, 8.292359007113204141662977783237, 9.255252371935305667217763298926, 10.44430963298397615603145384785, 10.88898669659775931858399105151, 11.99352188418744309507191814941

Graph of the ZZ-function along the critical line