Properties

Label 2-352-88.69-c1-0-1
Degree 22
Conductor 352352
Sign 0.3620.932i-0.362 - 0.932i
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  + (0.582 − 0.189i)3-s + (−0.858 + 1.18i)5-s + (−1.36 + 4.21i)7-s + (−2.12 + 1.54i)9-s + (−2.69 + 1.93i)11-s + (−3.27 − 4.50i)13-s + (−0.276 + 0.850i)15-s + (2.06 + 1.50i)17-s + (2.87 − 0.933i)19-s + 2.71i·21-s + 3.70·23-s + (0.886 + 2.72i)25-s + (−2.02 + 2.78i)27-s + (3.48 + 1.13i)29-s + (1.20 − 0.876i)31-s + ⋯
L(s)  = 1  + (0.336 − 0.109i)3-s + (−0.383 + 0.528i)5-s + (−0.517 + 1.59i)7-s + (−0.707 + 0.514i)9-s + (−0.811 + 0.584i)11-s + (−0.908 − 1.25i)13-s + (−0.0713 + 0.219i)15-s + (0.501 + 0.364i)17-s + (0.658 − 0.214i)19-s + 0.591i·21-s + 0.773·23-s + (0.177 + 0.545i)25-s + (−0.389 + 0.536i)27-s + (0.646 + 0.209i)29-s + (0.216 − 0.157i)31-s + ⋯

Functional equation

Λ(s)=(352s/2ΓC(s)L(s)=((0.3620.932i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.362 - 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(352s/2ΓC(s+1/2)L(s)=((0.3620.932i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.362 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 352352    =    25112^{5} \cdot 11
Sign: 0.3620.932i-0.362 - 0.932i
Analytic conductor: 2.810732.81073
Root analytic conductor: 1.676521.67652
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ352(113,)\chi_{352} (113, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 352, ( :1/2), 0.3620.932i)(2,\ 352,\ (\ :1/2),\ -0.362 - 0.932i)

Particular Values

L(1)L(1) \approx 0.538109+0.786521i0.538109 + 0.786521i
L(12)L(\frac12) \approx 0.538109+0.786521i0.538109 + 0.786521i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
11 1+(2.691.93i)T 1 + (2.69 - 1.93i)T
good3 1+(0.582+0.189i)T+(2.421.76i)T2 1 + (-0.582 + 0.189i)T + (2.42 - 1.76i)T^{2}
5 1+(0.8581.18i)T+(1.544.75i)T2 1 + (0.858 - 1.18i)T + (-1.54 - 4.75i)T^{2}
7 1+(1.364.21i)T+(5.664.11i)T2 1 + (1.36 - 4.21i)T + (-5.66 - 4.11i)T^{2}
13 1+(3.27+4.50i)T+(4.01+12.3i)T2 1 + (3.27 + 4.50i)T + (-4.01 + 12.3i)T^{2}
17 1+(2.061.50i)T+(5.25+16.1i)T2 1 + (-2.06 - 1.50i)T + (5.25 + 16.1i)T^{2}
19 1+(2.87+0.933i)T+(15.311.1i)T2 1 + (-2.87 + 0.933i)T + (15.3 - 11.1i)T^{2}
23 13.70T+23T2 1 - 3.70T + 23T^{2}
29 1+(3.481.13i)T+(23.4+17.0i)T2 1 + (-3.48 - 1.13i)T + (23.4 + 17.0i)T^{2}
31 1+(1.20+0.876i)T+(9.5729.4i)T2 1 + (-1.20 + 0.876i)T + (9.57 - 29.4i)T^{2}
37 1+(1.410.460i)T+(29.9+21.7i)T2 1 + (-1.41 - 0.460i)T + (29.9 + 21.7i)T^{2}
41 1+(0.9702.98i)T+(33.1+24.0i)T2 1 + (-0.970 - 2.98i)T + (-33.1 + 24.0i)T^{2}
43 1+6.25iT43T2 1 + 6.25iT - 43T^{2}
47 1+(0.8212.52i)T+(38.0+27.6i)T2 1 + (-0.821 - 2.52i)T + (-38.0 + 27.6i)T^{2}
53 1+(5.757.92i)T+(16.3+50.4i)T2 1 + (-5.75 - 7.92i)T + (-16.3 + 50.4i)T^{2}
59 1+(2.450.796i)T+(47.7+34.6i)T2 1 + (-2.45 - 0.796i)T + (47.7 + 34.6i)T^{2}
61 1+(3.284.51i)T+(18.858.0i)T2 1 + (3.28 - 4.51i)T + (-18.8 - 58.0i)T^{2}
67 111.5iT67T2 1 - 11.5iT - 67T^{2}
71 1+(0.9090.660i)T+(21.9+67.5i)T2 1 + (-0.909 - 0.660i)T + (21.9 + 67.5i)T^{2}
73 1+(4.36+13.4i)T+(59.042.9i)T2 1 + (-4.36 + 13.4i)T + (-59.0 - 42.9i)T^{2}
79 1+(0.619+0.450i)T+(24.475.1i)T2 1 + (-0.619 + 0.450i)T + (24.4 - 75.1i)T^{2}
83 1+(5.036.92i)T+(25.678.9i)T2 1 + (5.03 - 6.92i)T + (-25.6 - 78.9i)T^{2}
89 1+8.84T+89T2 1 + 8.84T + 89T^{2}
97 1+(0.241+0.175i)T+(29.992.2i)T2 1 + (-0.241 + 0.175i)T + (29.9 - 92.2i)T^{2}
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   L(s)=p j=12(1αj,pps)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

−11.90182850425433833496953353769, −10.81498168967777664590787163123, −9.916550581096086060846570344136, −8.913548855960224507622993697164, −7.993495003839717367719290363504, −7.22299053829820748943282944406, −5.72467065889602698625336004896, −5.13292582710670235986446925545, −3.00911810451099059063107542960, −2.61721445623282846499850796311, 0.61388405386499024225865285607, 2.93732452195337259853817671863, 4.00870037131012106789797311325, 5.06142786117801898677767509941, 6.54598665635207576652885652312, 7.45185210938260658276553600247, 8.352333335279250178355384992024, 9.424428668279162609352686214967, 10.14199365434962644369535468032, 11.23303357930251296268241409641

Graph of the ZZ-function along the critical line