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Properties

Label 2-352-88.51-c1-0-1
Degree 22
Conductor 352352
Sign −0.732−0.680i-0.732 - 0.680i
Analytic cond. 2.810732.81073
Root an. cond. 1.676521.67652
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  + (−0.852 + 2.62i)3-s + (1.35 + 1.86i)5-s + (1.08 + 3.35i)7-s + (−3.73 − 2.71i)9-s + (1.06 − 3.14i)11-s + (0.874 + 0.635i)13-s + (−6.04 + 1.96i)15-s + (0.128 + 0.176i)17-s + (−2.01 − 0.653i)19-s − 9.73·21-s − 5.68i·23-s + (−0.0920 + 0.283i)25-s + (3.60 − 2.61i)27-s + (2.14 + 6.60i)29-s + (−2.33 + 3.21i)31-s + ⋯
L(s)  = 1  + (−0.492 + 1.51i)3-s + (0.605 + 0.832i)5-s + (0.411 + 1.26i)7-s + (−1.24 − 0.903i)9-s + (0.321 − 0.946i)11-s + (0.242 + 0.176i)13-s + (−1.55 + 0.506i)15-s + (0.0311 + 0.0428i)17-s + (−0.461 − 0.149i)19-s − 2.12·21-s − 1.18i·23-s + (−0.0184 + 0.0566i)25-s + (0.693 − 0.503i)27-s + (0.398 + 1.22i)29-s + (−0.419 + 0.577i)31-s + ⋯

Functional equation

Λ(s)=(352s/2 ΓC(s) L(s)=((−0.732−0.680i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.732 - 0.680i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(352s/2 ΓC(s+1/2) L(s)=((−0.732−0.680i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.732 - 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 352352    =    25⋅112^{5} \cdot 11
Sign: −0.732−0.680i-0.732 - 0.680i
Analytic conductor: 2.810732.81073
Root analytic conductor: 1.676521.67652
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ352(271,⋅)\chi_{352} (271, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 352, ( :1/2), −0.732−0.680i)(2,\ 352,\ (\ :1/2),\ -0.732 - 0.680i)

Particular Values

L(1)L(1) ≈\approx 0.458699+1.16690i0.458699 + 1.16690i
L(12)L(\frac12) ≈\approx 0.458699+1.16690i0.458699 + 1.16690i
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
11 1+(−1.06+3.14i)T 1 + (-1.06 + 3.14i)T
good3 1+(0.852−2.62i)T+(−2.42−1.76i)T2 1 + (0.852 - 2.62i)T + (-2.42 - 1.76i)T^{2}
5 1+(−1.35−1.86i)T+(−1.54+4.75i)T2 1 + (-1.35 - 1.86i)T + (-1.54 + 4.75i)T^{2}
7 1+(−1.08−3.35i)T+(−5.66+4.11i)T2 1 + (-1.08 - 3.35i)T + (-5.66 + 4.11i)T^{2}
13 1+(−0.874−0.635i)T+(4.01+12.3i)T2 1 + (-0.874 - 0.635i)T + (4.01 + 12.3i)T^{2}
17 1+(−0.128−0.176i)T+(−5.25+16.1i)T2 1 + (-0.128 - 0.176i)T + (-5.25 + 16.1i)T^{2}
19 1+(2.01+0.653i)T+(15.3+11.1i)T2 1 + (2.01 + 0.653i)T + (15.3 + 11.1i)T^{2}
23 1+5.68iT−23T2 1 + 5.68iT - 23T^{2}
29 1+(−2.14−6.60i)T+(−23.4+17.0i)T2 1 + (-2.14 - 6.60i)T + (-23.4 + 17.0i)T^{2}
31 1+(2.33−3.21i)T+(−9.57−29.4i)T2 1 + (2.33 - 3.21i)T + (-9.57 - 29.4i)T^{2}
37 1+(2.44−0.794i)T+(29.9−21.7i)T2 1 + (2.44 - 0.794i)T + (29.9 - 21.7i)T^{2}
41 1+(3.41+1.10i)T+(33.1+24.0i)T2 1 + (3.41 + 1.10i)T + (33.1 + 24.0i)T^{2}
43 1+9.45iT−43T2 1 + 9.45iT - 43T^{2}
47 1+(−12.1−3.93i)T+(38.0+27.6i)T2 1 + (-12.1 - 3.93i)T + (38.0 + 27.6i)T^{2}
53 1+(−5.28+7.27i)T+(−16.3−50.4i)T2 1 + (-5.28 + 7.27i)T + (-16.3 - 50.4i)T^{2}
59 1+(−1.96−6.03i)T+(−47.7+34.6i)T2 1 + (-1.96 - 6.03i)T + (-47.7 + 34.6i)T^{2}
61 1+(2.23−1.62i)T+(18.8−58.0i)T2 1 + (2.23 - 1.62i)T + (18.8 - 58.0i)T^{2}
67 1−6.19T+67T2 1 - 6.19T + 67T^{2}
71 1+(−1.31−1.81i)T+(−21.9+67.5i)T2 1 + (-1.31 - 1.81i)T + (-21.9 + 67.5i)T^{2}
73 1+(1.71−0.557i)T+(59.0−42.9i)T2 1 + (1.71 - 0.557i)T + (59.0 - 42.9i)T^{2}
79 1+(0.386+0.280i)T+(24.4+75.1i)T2 1 + (0.386 + 0.280i)T + (24.4 + 75.1i)T^{2}
83 1+(−2.12−2.92i)T+(−25.6+78.9i)T2 1 + (-2.12 - 2.92i)T + (-25.6 + 78.9i)T^{2}
89 1−8.99T+89T2 1 - 8.99T + 89T^{2}
97 1+(−5.99−4.35i)T+(29.9+92.2i)T2 1 + (-5.99 - 4.35i)T + (29.9 + 92.2i)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

−11.55059464142450271476511758855, −10.66473806484909880050839292907, −10.32263169266891216871878419890, −8.928703660020019011375248979653, −8.717062123342835198203789044036, −6.68714279065770331131764282792, −5.79934567126743077430563950636, −5.06105809115120818490714871526, −3.72486666570824065450792219133, −2.51773465552381306163268310117, 0.998440747121527647781988813123, 1.93393396180261147973891559175, 4.17399131242095572365961141552, 5.38364266023255405072822230281, 6.40599626292678851261569558554, 7.36130183497546521746571504807, 7.941329183843956647287546810882, 9.246232487014499894759913458091, 10.28265903745197167082482914465, 11.36639351398924816760187698343

Graph of the ZZ-function along the critical line