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Properties

Label 2-55-11.9-c1-0-2
Degree 22
Conductor 5555
Sign 0.977+0.209i0.977 + 0.209i
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  + (0.147 − 0.453i)2-s + (−0.261 + 0.189i)3-s + (1.43 + 1.04i)4-s + (−0.309 − 0.951i)5-s + (0.0476 + 0.146i)6-s + (−2.17 − 1.57i)7-s + (1.45 − 1.05i)8-s + (−0.894 + 2.75i)9-s − 0.477·10-s + (−2.79 − 1.79i)11-s − 0.572·12-s + (−1.44 + 4.43i)13-s + (−1.03 + 0.753i)14-s + (0.261 + 0.189i)15-s + (0.829 + 2.55i)16-s + (−1.42 − 4.39i)17-s + ⋯
L(s)  = 1  + (0.104 − 0.320i)2-s + (−0.150 + 0.109i)3-s + (0.716 + 0.520i)4-s + (−0.138 − 0.425i)5-s + (0.0194 + 0.0598i)6-s + (−0.821 − 0.596i)7-s + (0.514 − 0.374i)8-s + (−0.298 + 0.917i)9-s − 0.150·10-s + (−0.841 − 0.540i)11-s − 0.165·12-s + (−0.400 + 1.23i)13-s + (−0.277 + 0.201i)14-s + (0.0674 + 0.0490i)15-s + (0.207 + 0.638i)16-s + (−0.346 − 1.06i)17-s + ⋯

Functional equation

Λ(s)=(55s/2 ΓC(s) L(s)=((0.977+0.209i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.209i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(55s/2 ΓC(s+1/2) L(s)=((0.977+0.209i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 + 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5555    =    5⋅115 \cdot 11
Sign: 0.977+0.209i0.977 + 0.209i
Analytic conductor: 0.4391770.439177
Root analytic conductor: 0.6627040.662704
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ55(31,⋅)\chi_{55} (31, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 55, ( :1/2), 0.977+0.209i)(2,\ 55,\ (\ :1/2),\ 0.977 + 0.209i)

Particular Values

L(1)L(1) ≈\approx 0.888300−0.0941154i0.888300 - 0.0941154i
L(12)L(\frac12) ≈\approx 0.888300−0.0941154i0.888300 - 0.0941154i
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+(2.79+1.79i)T 1 + (2.79 + 1.79i)T
good2 1+(−0.147+0.453i)T+(−1.61−1.17i)T2 1 + (-0.147 + 0.453i)T + (-1.61 - 1.17i)T^{2}
3 1+(0.261−0.189i)T+(0.927−2.85i)T2 1 + (0.261 - 0.189i)T + (0.927 - 2.85i)T^{2}
7 1+(2.17+1.57i)T+(2.16+6.65i)T2 1 + (2.17 + 1.57i)T + (2.16 + 6.65i)T^{2}
13 1+(1.44−4.43i)T+(−10.5−7.64i)T2 1 + (1.44 - 4.43i)T + (-10.5 - 7.64i)T^{2}
17 1+(1.42+4.39i)T+(−13.7+9.99i)T2 1 + (1.42 + 4.39i)T + (-13.7 + 9.99i)T^{2}
19 1+(−3.51+2.55i)T+(5.87−18.0i)T2 1 + (-3.51 + 2.55i)T + (5.87 - 18.0i)T^{2}
23 1−2.77T+23T2 1 - 2.77T + 23T^{2}
29 1+(−2.43−1.77i)T+(8.96+27.5i)T2 1 + (-2.43 - 1.77i)T + (8.96 + 27.5i)T^{2}
31 1+(−0.737+2.26i)T+(−25.0−18.2i)T2 1 + (-0.737 + 2.26i)T + (-25.0 - 18.2i)T^{2}
37 1+(−8.61−6.25i)T+(11.4+35.1i)T2 1 + (-8.61 - 6.25i)T + (11.4 + 35.1i)T^{2}
41 1+(−1.78+1.29i)T+(12.6−38.9i)T2 1 + (-1.78 + 1.29i)T + (12.6 - 38.9i)T^{2}
43 1+7.06T+43T2 1 + 7.06T + 43T^{2}
47 1+(3.52−2.56i)T+(14.5−44.6i)T2 1 + (3.52 - 2.56i)T + (14.5 - 44.6i)T^{2}
53 1+(1.95−6.02i)T+(−42.8−31.1i)T2 1 + (1.95 - 6.02i)T + (-42.8 - 31.1i)T^{2}
59 1+(9.50+6.90i)T+(18.2+56.1i)T2 1 + (9.50 + 6.90i)T + (18.2 + 56.1i)T^{2}
61 1+(1.23+3.78i)T+(−49.3+35.8i)T2 1 + (1.23 + 3.78i)T + (-49.3 + 35.8i)T^{2}
67 1−7.31T+67T2 1 - 7.31T + 67T^{2}
71 1+(−0.369−1.13i)T+(−57.4+41.7i)T2 1 + (-0.369 - 1.13i)T + (-57.4 + 41.7i)T^{2}
73 1+(0.826+0.600i)T+(22.5+69.4i)T2 1 + (0.826 + 0.600i)T + (22.5 + 69.4i)T^{2}
79 1+(−1.08+3.33i)T+(−63.9−46.4i)T2 1 + (-1.08 + 3.33i)T + (-63.9 - 46.4i)T^{2}
83 1+(−3.43−10.5i)T+(−67.1+48.7i)T2 1 + (-3.43 - 10.5i)T + (-67.1 + 48.7i)T^{2}
89 1−2.76T+89T2 1 - 2.76T + 89T^{2}
97 1+(−5.72+17.6i)T+(−78.4−57.0i)T2 1 + (-5.72 + 17.6i)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

−15.75056155440062040278034266744, −13.77825355802615383516846051634, −13.10901603888172762826625632825, −11.72468626707099500079624711830, −10.94055486123167482789326732604, −9.566302831696967268366743321529, −7.896252055725894899091818898829, −6.75492738464691154106956630459, −4.77179832299571594616882858543, −2.86878193548490114280889880405, 2.92168491018701487635908797969, 5.55342796449956502906124677934, 6.52183756843724344126252072339, 7.85046637762129686202667974854, 9.708002093866407336057764557194, 10.68483016970310806188454072607, 12.04389045753621286005671996665, 12.99109177053680068805776304200, 14.76110512962494513799275434135, 15.24017275597784785293328193792

Graph of the ZZ-function along the critical line