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Properties

Label 2-320-16.5-c1-0-4
Degree 22
Conductor 320320
Sign 0.802+0.596i0.802 + 0.596i
Analytic cond. 2.555212.55521
Root an. cond. 1.598501.59850
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  + (−1.82 + 1.82i)3-s + (−0.707 − 0.707i)5-s − 4.50i·7-s − 3.68i·9-s + (1.64 + 1.64i)11-s + (1.51 − 1.51i)13-s + 2.58·15-s + 1.45·17-s + (2.67 − 2.67i)19-s + (8.24 + 8.24i)21-s − 2.37i·23-s + 1.00i·25-s + (1.24 + 1.24i)27-s + (0.924 − 0.924i)29-s + 7.20·31-s + ⋯
L(s)  = 1  + (−1.05 + 1.05i)3-s + (−0.316 − 0.316i)5-s − 1.70i·7-s − 1.22i·9-s + (0.494 + 0.494i)11-s + (0.421 − 0.421i)13-s + 0.667·15-s + 0.353·17-s + (0.614 − 0.614i)19-s + (1.79 + 1.79i)21-s − 0.495i·23-s + 0.200i·25-s + (0.239 + 0.239i)27-s + (0.171 − 0.171i)29-s + 1.29·31-s + ⋯

Functional equation

Λ(s)=(320s/2 ΓC(s) L(s)=((0.802+0.596i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.802 + 0.596i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(320s/2 ΓC(s+1/2) L(s)=((0.802+0.596i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.802 + 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 320320    =    26⋅52^{6} \cdot 5
Sign: 0.802+0.596i0.802 + 0.596i
Analytic conductor: 2.555212.55521
Root analytic conductor: 1.598501.59850
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ320(241,⋅)\chi_{320} (241, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 320, ( :1/2), 0.802+0.596i)(2,\ 320,\ (\ :1/2),\ 0.802 + 0.596i)

Particular Values

L(1)L(1) ≈\approx 0.782562−0.258700i0.782562 - 0.258700i
L(12)L(\frac12) ≈\approx 0.782562−0.258700i0.782562 - 0.258700i
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
5 1+(0.707+0.707i)T 1 + (0.707 + 0.707i)T
good3 1+(1.82−1.82i)T−3iT2 1 + (1.82 - 1.82i)T - 3iT^{2}
7 1+4.50iT−7T2 1 + 4.50iT - 7T^{2}
11 1+(−1.64−1.64i)T+11iT2 1 + (-1.64 - 1.64i)T + 11iT^{2}
13 1+(−1.51+1.51i)T−13iT2 1 + (-1.51 + 1.51i)T - 13iT^{2}
17 1−1.45T+17T2 1 - 1.45T + 17T^{2}
19 1+(−2.67+2.67i)T−19iT2 1 + (-2.67 + 2.67i)T - 19iT^{2}
23 1+2.37iT−23T2 1 + 2.37iT - 23T^{2}
29 1+(−0.924+0.924i)T−29iT2 1 + (-0.924 + 0.924i)T - 29iT^{2}
31 1−7.20T+31T2 1 - 7.20T + 31T^{2}
37 1+(5.21+5.21i)T+37iT2 1 + (5.21 + 5.21i)T + 37iT^{2}
41 1+6.41iT−41T2 1 + 6.41iT - 41T^{2}
43 1+(7.65+7.65i)T+43iT2 1 + (7.65 + 7.65i)T + 43iT^{2}
47 1−2.51T+47T2 1 - 2.51T + 47T^{2}
53 1+(−1.50−1.50i)T+53iT2 1 + (-1.50 - 1.50i)T + 53iT^{2}
59 1+(−5.31−5.31i)T+59iT2 1 + (-5.31 - 5.31i)T + 59iT^{2}
61 1+(1.02−1.02i)T−61iT2 1 + (1.02 - 1.02i)T - 61iT^{2}
67 1+(5.22−5.22i)T−67iT2 1 + (5.22 - 5.22i)T - 67iT^{2}
71 1−1.92iT−71T2 1 - 1.92iT - 71T^{2}
73 1+1.39iT−73T2 1 + 1.39iT - 73T^{2}
79 1+5.06T+79T2 1 + 5.06T + 79T^{2}
83 1+(−2.44+2.44i)T−83iT2 1 + (-2.44 + 2.44i)T - 83iT^{2}
89 1−9.36iT−89T2 1 - 9.36iT - 89T^{2}
97 1−18.6T+97T2 1 - 18.6T + 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

−11.42450216439130688724268164666, −10.37259312778410134648946872006, −10.20965755795330654867934254382, −8.913509433496339327336597777378, −7.55647576418831794117671529252, −6.64881078138693082832598768608, −5.32552306363575352672770828373, −4.40065229662738916320831859637, −3.68491556955770368355681040122, −0.74909416895864156142606767649, 1.53357315512767833986895715953, 3.15258777974236311583714727115, 5.08458029858295570157388670468, 6.05942768389190035610061317668, 6.56218749407795175855501776041, 7.86662047887083094400384534470, 8.737088699249460166958497436345, 9.930050969885592271306410018917, 11.39655398438151749218490339524, 11.71172315517903076713156378396

Graph of the ZZ-function along the critical line