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Properties

Label 2-55-11.5-c1-0-0
Degree 22
Conductor 5555
Sign 0.0851−0.996i0.0851 - 0.996i
Analytic cond. 0.4391770.439177
Root an. cond. 0.6627040.662704
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  + (0.697 + 2.14i)2-s + (−0.628 − 0.456i)3-s + (−2.50 + 1.82i)4-s + (0.309 − 0.951i)5-s + (0.542 − 1.66i)6-s + (−0.100 + 0.0728i)7-s + (−2.00 − 1.45i)8-s + (−0.740 − 2.27i)9-s + 2.25·10-s + (−0.899 − 3.19i)11-s + 2.40·12-s + (1.69 + 5.22i)13-s + (−0.226 − 0.164i)14-s + (−0.628 + 0.456i)15-s + (−0.184 + 0.566i)16-s + (0.160 − 0.494i)17-s + ⋯
L(s)  = 1  + (0.493 + 1.51i)2-s + (−0.363 − 0.263i)3-s + (−1.25 + 0.910i)4-s + (0.138 − 0.425i)5-s + (0.221 − 0.681i)6-s + (−0.0379 + 0.0275i)7-s + (−0.709 − 0.515i)8-s + (−0.246 − 0.759i)9-s + 0.714·10-s + (−0.271 − 0.962i)11-s + 0.695·12-s + (0.470 + 1.44i)13-s + (−0.0605 − 0.0439i)14-s + (−0.162 + 0.117i)15-s + (−0.0460 + 0.141i)16-s + (0.0389 − 0.119i)17-s + ⋯

Functional equation

Λ(s)=(55s/2 ΓC(s) L(s)=((0.0851−0.996i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0851 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(55s/2 ΓC(s+1/2) L(s)=((0.0851−0.996i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0851 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5555    =    5⋅115 \cdot 11
Sign: 0.0851−0.996i0.0851 - 0.996i
Analytic conductor: 0.4391770.439177
Root analytic conductor: 0.6627040.662704
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ55(16,⋅)\chi_{55} (16, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 55, ( :1/2), 0.0851−0.996i)(2,\ 55,\ (\ :1/2),\ 0.0851 - 0.996i)

Particular Values

L(1)L(1) ≈\approx 0.690106+0.633675i0.690106 + 0.633675i
L(12)L(\frac12) ≈\approx 0.690106+0.633675i0.690106 + 0.633675i
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)
bad5 1+(−0.309+0.951i)T 1 + (-0.309 + 0.951i)T
11 1+(0.899+3.19i)T 1 + (0.899 + 3.19i)T
good2 1+(−0.697−2.14i)T+(−1.61+1.17i)T2 1 + (-0.697 - 2.14i)T + (-1.61 + 1.17i)T^{2}
3 1+(0.628+0.456i)T+(0.927+2.85i)T2 1 + (0.628 + 0.456i)T + (0.927 + 2.85i)T^{2}
7 1+(0.100−0.0728i)T+(2.16−6.65i)T2 1 + (0.100 - 0.0728i)T + (2.16 - 6.65i)T^{2}
13 1+(−1.69−5.22i)T+(−10.5+7.64i)T2 1 + (-1.69 - 5.22i)T + (-10.5 + 7.64i)T^{2}
17 1+(−0.160+0.494i)T+(−13.7−9.99i)T2 1 + (-0.160 + 0.494i)T + (-13.7 - 9.99i)T^{2}
19 1+(2.55+1.85i)T+(5.87+18.0i)T2 1 + (2.55 + 1.85i)T + (5.87 + 18.0i)T^{2}
23 1+7.92T+23T2 1 + 7.92T + 23T^{2}
29 1+(−3.29+2.39i)T+(8.96−27.5i)T2 1 + (-3.29 + 2.39i)T + (8.96 - 27.5i)T^{2}
31 1+(−2.17−6.70i)T+(−25.0+18.2i)T2 1 + (-2.17 - 6.70i)T + (-25.0 + 18.2i)T^{2}
37 1+(−7.10+5.16i)T+(11.4−35.1i)T2 1 + (-7.10 + 5.16i)T + (11.4 - 35.1i)T^{2}
41 1+(−6.10−4.43i)T+(12.6+38.9i)T2 1 + (-6.10 - 4.43i)T + (12.6 + 38.9i)T^{2}
43 1−3.42T+43T2 1 - 3.42T + 43T^{2}
47 1+(0.369+0.268i)T+(14.5+44.6i)T2 1 + (0.369 + 0.268i)T + (14.5 + 44.6i)T^{2}
53 1+(−0.0109−0.0337i)T+(−42.8+31.1i)T2 1 + (-0.0109 - 0.0337i)T + (-42.8 + 31.1i)T^{2}
59 1+(−4.42+3.21i)T+(18.2−56.1i)T2 1 + (-4.42 + 3.21i)T + (18.2 - 56.1i)T^{2}
61 1+(−2.37+7.31i)T+(−49.3−35.8i)T2 1 + (-2.37 + 7.31i)T + (-49.3 - 35.8i)T^{2}
67 1+2.53T+67T2 1 + 2.53T + 67T^{2}
71 1+(−3.79+11.6i)T+(−57.4−41.7i)T2 1 + (-3.79 + 11.6i)T + (-57.4 - 41.7i)T^{2}
73 1+(6.89−5.00i)T+(22.5−69.4i)T2 1 + (6.89 - 5.00i)T + (22.5 - 69.4i)T^{2}
79 1+(−1.93−5.96i)T+(−63.9+46.4i)T2 1 + (-1.93 - 5.96i)T + (-63.9 + 46.4i)T^{2}
83 1+(−0.193+0.595i)T+(−67.1−48.7i)T2 1 + (-0.193 + 0.595i)T + (-67.1 - 48.7i)T^{2}
89 1+10.1T+89T2 1 + 10.1T + 89T^{2}
97 1+(0.567+1.74i)T+(−78.4+57.0i)T2 1 + (0.567 + 1.74i)T + (-78.4 + 57.0i)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

−15.86939656166084206603023070761, −14.41569646073921159194903023922, −13.75862915244577407546812506468, −12.53987093821885259938117135949, −11.26966056705669241131804940693, −9.195103899906610841084658852914, −8.156149061184127198034574043138, −6.57593427990265695386227359779, −5.88842260242961702141504186232, −4.26370428509084885869492079320, 2.46353696021421585871651661305, 4.22664295371424507929169692566, 5.71379921997527799409118307459, 7.937053824564291706663725917732, 10.03781571287046251319591629114, 10.45175197486914553598723219502, 11.57008253435964467006764239703, 12.68468734144618171619578504779, 13.55743276629324223939653622966, 14.78526946303714637514524313179

Graph of the ZZ-function along the critical line