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Properties

Label 2-352-11.3-c1-0-4
Degree 22
Conductor 352352
Sign 0.878+0.477i0.878 + 0.477i
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  + (−0.212 − 0.653i)3-s + (−0.607 − 0.441i)5-s + (−0.944 + 2.90i)7-s + (2.04 − 1.48i)9-s + (3.27 − 0.510i)11-s + (5.02 − 3.64i)13-s + (−0.159 + 0.490i)15-s + (2.49 + 1.81i)17-s + (−1.30 − 4.02i)19-s + 2.09·21-s + 0.264·23-s + (−1.37 − 4.21i)25-s + (−3.07 − 2.23i)27-s + (−0.513 + 1.57i)29-s + (−3.39 + 2.46i)31-s + ⋯
L(s)  = 1  + (−0.122 − 0.377i)3-s + (−0.271 − 0.197i)5-s + (−0.356 + 1.09i)7-s + (0.681 − 0.495i)9-s + (0.988 − 0.153i)11-s + (1.39 − 1.01i)13-s + (−0.0411 + 0.126i)15-s + (0.604 + 0.439i)17-s + (−0.300 − 0.924i)19-s + 0.457·21-s + 0.0551·23-s + (−0.274 − 0.843i)25-s + (−0.591 − 0.429i)27-s + (−0.0952 + 0.293i)29-s + (−0.609 + 0.443i)31-s + ⋯

Functional equation

Λ(s)=(352s/2 ΓC(s) L(s)=((0.878+0.477i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.878 + 0.477i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(352s/2 ΓC(s+1/2) L(s)=((0.878+0.477i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.878 + 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 352352    =    25⋅112^{5} \cdot 11
Sign: 0.878+0.477i0.878 + 0.477i
Analytic conductor: 2.810732.81073
Root analytic conductor: 1.676521.67652
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ352(289,⋅)\chi_{352} (289, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 352, ( :1/2), 0.878+0.477i)(2,\ 352,\ (\ :1/2),\ 0.878 + 0.477i)

Particular Values

L(1)L(1) ≈\approx 1.30638−0.331976i1.30638 - 0.331976i
L(12)L(\frac12) ≈\approx 1.30638−0.331976i1.30638 - 0.331976i
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+(−3.27+0.510i)T 1 + (-3.27 + 0.510i)T
good3 1+(0.212+0.653i)T+(−2.42+1.76i)T2 1 + (0.212 + 0.653i)T + (-2.42 + 1.76i)T^{2}
5 1+(0.607+0.441i)T+(1.54+4.75i)T2 1 + (0.607 + 0.441i)T + (1.54 + 4.75i)T^{2}
7 1+(0.944−2.90i)T+(−5.66−4.11i)T2 1 + (0.944 - 2.90i)T + (-5.66 - 4.11i)T^{2}
13 1+(−5.02+3.64i)T+(4.01−12.3i)T2 1 + (-5.02 + 3.64i)T + (4.01 - 12.3i)T^{2}
17 1+(−2.49−1.81i)T+(5.25+16.1i)T2 1 + (-2.49 - 1.81i)T + (5.25 + 16.1i)T^{2}
19 1+(1.30+4.02i)T+(−15.3+11.1i)T2 1 + (1.30 + 4.02i)T + (-15.3 + 11.1i)T^{2}
23 1−0.264T+23T2 1 - 0.264T + 23T^{2}
29 1+(0.513−1.57i)T+(−23.4−17.0i)T2 1 + (0.513 - 1.57i)T + (-23.4 - 17.0i)T^{2}
31 1+(3.39−2.46i)T+(9.57−29.4i)T2 1 + (3.39 - 2.46i)T + (9.57 - 29.4i)T^{2}
37 1+(0.730−2.24i)T+(−29.9−21.7i)T2 1 + (0.730 - 2.24i)T + (-29.9 - 21.7i)T^{2}
41 1+(−1.75−5.40i)T+(−33.1+24.0i)T2 1 + (-1.75 - 5.40i)T + (-33.1 + 24.0i)T^{2}
43 1−9.90T+43T2 1 - 9.90T + 43T^{2}
47 1+(−3.56−10.9i)T+(−38.0+27.6i)T2 1 + (-3.56 - 10.9i)T + (-38.0 + 27.6i)T^{2}
53 1+(3.56−2.59i)T+(16.3−50.4i)T2 1 + (3.56 - 2.59i)T + (16.3 - 50.4i)T^{2}
59 1+(−0.258+0.795i)T+(−47.7−34.6i)T2 1 + (-0.258 + 0.795i)T + (-47.7 - 34.6i)T^{2}
61 1+(5.06+3.68i)T+(18.8+58.0i)T2 1 + (5.06 + 3.68i)T + (18.8 + 58.0i)T^{2}
67 1+7.76T+67T2 1 + 7.76T + 67T^{2}
71 1+(8.38+6.08i)T+(21.9+67.5i)T2 1 + (8.38 + 6.08i)T + (21.9 + 67.5i)T^{2}
73 1+(2.61−8.05i)T+(−59.0−42.9i)T2 1 + (2.61 - 8.05i)T + (-59.0 - 42.9i)T^{2}
79 1+(−11.9+8.65i)T+(24.4−75.1i)T2 1 + (-11.9 + 8.65i)T + (24.4 - 75.1i)T^{2}
83 1+(8.89+6.46i)T+(25.6+78.9i)T2 1 + (8.89 + 6.46i)T + (25.6 + 78.9i)T^{2}
89 1+15.9T+89T2 1 + 15.9T + 89T^{2}
97 1+(−0.712+0.518i)T+(29.9−92.2i)T2 1 + (-0.712 + 0.518i)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

−11.53564141666253845845202093800, −10.58614212346309099754543108698, −9.334444958011983244020150456066, −8.726414480271267955617050968484, −7.66625406446278572715803017969, −6.36054543764097806363802591218, −5.86618623738369048595353038582, −4.25672846640901186254303435613, −3.08086300463060817169677127945, −1.22652799447617814410504899633, 1.49727122205353816728122679298, 3.86932828898630329915412898891, 4.03316463536418855775401602707, 5.74972688702494134816937275467, 6.91329608172213728943230432962, 7.56425139784204039372077840409, 8.932863132896721281724733392574, 9.792629330716071804561919765323, 10.68247512711097345909767562619, 11.34085069178250904962233246928

Graph of the ZZ-function along the critical line