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Properties

Label 2-352-88.37-c1-0-1
Degree 22
Conductor 352352
Sign 0.159−0.987i0.159 - 0.987i
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  + (−0.826 − 0.268i)3-s + (2.15 + 2.97i)5-s + (0.369 + 1.13i)7-s + (−1.81 − 1.31i)9-s + (−2.03 + 2.62i)11-s + (−2.33 + 3.22i)13-s + (−0.986 − 3.03i)15-s + (2.57 − 1.86i)17-s + (4.09 + 1.33i)19-s − 1.03i·21-s − 2.46·23-s + (−2.62 + 8.08i)25-s + (2.67 + 3.68i)27-s + (−3.65 + 1.18i)29-s + (7.40 + 5.37i)31-s + ⋯
L(s)  = 1  + (−0.476 − 0.154i)3-s + (0.965 + 1.32i)5-s + (0.139 + 0.429i)7-s + (−0.605 − 0.439i)9-s + (−0.612 + 0.790i)11-s + (−0.648 + 0.893i)13-s + (−0.254 − 0.783i)15-s + (0.623 − 0.453i)17-s + (0.939 + 0.305i)19-s − 0.226i·21-s − 0.513·23-s + (−0.525 + 1.61i)25-s + (0.515 + 0.709i)27-s + (−0.678 + 0.220i)29-s + (1.32 + 0.965i)31-s + ⋯

Functional equation

Λ(s)=(352s/2 ΓC(s) L(s)=((0.159−0.987i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.159 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(352s/2 ΓC(s+1/2) L(s)=((0.159−0.987i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.159 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 352352    =    25⋅112^{5} \cdot 11
Sign: 0.159−0.987i0.159 - 0.987i
Analytic conductor: 2.810732.81073
Root analytic conductor: 1.676521.67652
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ352(81,⋅)\chi_{352} (81, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 352, ( :1/2), 0.159−0.987i)(2,\ 352,\ (\ :1/2),\ 0.159 - 0.987i)

Particular Values

L(1)L(1) ≈\approx 0.886780+0.754899i0.886780 + 0.754899i
L(12)L(\frac12) ≈\approx 0.886780+0.754899i0.886780 + 0.754899i
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+(2.03−2.62i)T 1 + (2.03 - 2.62i)T
good3 1+(0.826+0.268i)T+(2.42+1.76i)T2 1 + (0.826 + 0.268i)T + (2.42 + 1.76i)T^{2}
5 1+(−2.15−2.97i)T+(−1.54+4.75i)T2 1 + (-2.15 - 2.97i)T + (-1.54 + 4.75i)T^{2}
7 1+(−0.369−1.13i)T+(−5.66+4.11i)T2 1 + (-0.369 - 1.13i)T + (-5.66 + 4.11i)T^{2}
13 1+(2.33−3.22i)T+(−4.01−12.3i)T2 1 + (2.33 - 3.22i)T + (-4.01 - 12.3i)T^{2}
17 1+(−2.57+1.86i)T+(5.25−16.1i)T2 1 + (-2.57 + 1.86i)T + (5.25 - 16.1i)T^{2}
19 1+(−4.09−1.33i)T+(15.3+11.1i)T2 1 + (-4.09 - 1.33i)T + (15.3 + 11.1i)T^{2}
23 1+2.46T+23T2 1 + 2.46T + 23T^{2}
29 1+(3.65−1.18i)T+(23.4−17.0i)T2 1 + (3.65 - 1.18i)T + (23.4 - 17.0i)T^{2}
31 1+(−7.40−5.37i)T+(9.57+29.4i)T2 1 + (-7.40 - 5.37i)T + (9.57 + 29.4i)T^{2}
37 1+(−5.06+1.64i)T+(29.9−21.7i)T2 1 + (-5.06 + 1.64i)T + (29.9 - 21.7i)T^{2}
41 1+(−0.164+0.506i)T+(−33.1−24.0i)T2 1 + (-0.164 + 0.506i)T + (-33.1 - 24.0i)T^{2}
43 1+6.51iT−43T2 1 + 6.51iT - 43T^{2}
47 1+(0.110−0.341i)T+(−38.0−27.6i)T2 1 + (0.110 - 0.341i)T + (-38.0 - 27.6i)T^{2}
53 1+(1.42−1.96i)T+(−16.3−50.4i)T2 1 + (1.42 - 1.96i)T + (-16.3 - 50.4i)T^{2}
59 1+(−0.566+0.184i)T+(47.7−34.6i)T2 1 + (-0.566 + 0.184i)T + (47.7 - 34.6i)T^{2}
61 1+(4.28+5.89i)T+(−18.8+58.0i)T2 1 + (4.28 + 5.89i)T + (-18.8 + 58.0i)T^{2}
67 1+11.2iT−67T2 1 + 11.2iT - 67T^{2}
71 1+(−4.52+3.28i)T+(21.9−67.5i)T2 1 + (-4.52 + 3.28i)T + (21.9 - 67.5i)T^{2}
73 1+(0.825+2.54i)T+(−59.0+42.9i)T2 1 + (0.825 + 2.54i)T + (-59.0 + 42.9i)T^{2}
79 1+(9.30+6.75i)T+(24.4+75.1i)T2 1 + (9.30 + 6.75i)T + (24.4 + 75.1i)T^{2}
83 1+(0.589+0.811i)T+(−25.6+78.9i)T2 1 + (0.589 + 0.811i)T + (-25.6 + 78.9i)T^{2}
89 1−13.6T+89T2 1 - 13.6T + 89T^{2}
97 1+(−4.81−3.49i)T+(29.9+92.2i)T2 1 + (-4.81 - 3.49i)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

−11.77668482850698527358798941386, −10.71233951754887431392562277033, −9.910748952801114186415191941780, −9.210079646457421144460554831018, −7.64051335105587768244007967263, −6.80174137513423330687492737621, −5.93388084092272626962903803192, −5.04807496960307422197684502372, −3.15469731509560581993470434747, −2.10984562787973340211577485218, 0.867215513219699449907121849641, 2.70134427583386621394985670301, 4.54849520290496451742288363042, 5.51440889773300685885639279256, 5.89619034374636970156100213056, 7.77911507691093825161251959872, 8.373947843932854325231776451208, 9.620156892064580325935829418368, 10.22175093346658483844458698267, 11.27280711541998825507960024852

Graph of the ZZ-function along the critical line