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Properties

Label 2-11-11.10-c8-0-3
Degree 22
Conductor 1111
Sign 0.332−0.943i0.332 - 0.943i
Analytic cond. 4.481164.48116
Root an. cond. 2.116872.11687
Motivic weight 88
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 14.0i·2-s + 84.5·3-s + 59.3·4-s + 63.0·5-s + 1.18e3i·6-s + 1.66e3i·7-s + 4.42e3i·8-s + 582.·9-s + 884. i·10-s + (4.86e3 − 1.38e4i)11-s + 5.01e3·12-s − 1.61e4i·13-s − 2.33e4·14-s + 5.32e3·15-s − 4.68e4·16-s − 5.82e4i·17-s + ⋯
L(s)  = 1  + 0.876i·2-s + 1.04·3-s + 0.231·4-s + 0.100·5-s + 0.914i·6-s + 0.694i·7-s + 1.07i·8-s + 0.0887·9-s + 0.0884i·10-s + (0.332 − 0.943i)11-s + 0.241·12-s − 0.566i·13-s − 0.608·14-s + 0.105·15-s − 0.714·16-s − 0.697i·17-s + ⋯

Functional equation

Λ(s)=(11s/2 ΓC(s) L(s)=((0.332−0.943i) Λ‾(9−s)\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.332 - 0.943i)\, \overline{\Lambda}(9-s) \end{aligned}
Λ(s)=(11s/2 ΓC(s+4) L(s)=((0.332−0.943i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.332 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1111
Sign: 0.332−0.943i0.332 - 0.943i
Analytic conductor: 4.481164.48116
Root analytic conductor: 2.116872.11687
Motivic weight: 88
Rational: no
Arithmetic: yes
Character: χ11(10,⋅)\chi_{11} (10, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 11, ( :4), 0.332−0.943i)(2,\ 11,\ (\ :4),\ 0.332 - 0.943i)

Particular Values

L(92)L(\frac{9}{2}) ≈\approx 1.78911+1.26656i1.78911 + 1.26656i
L(12)L(\frac12) ≈\approx 1.78911+1.26656i1.78911 + 1.26656i
L(5)L(5) 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)
bad11 1+(−4.86e3+1.38e4i)T 1 + (-4.86e3 + 1.38e4i)T
good2 1−14.0iT−256T2 1 - 14.0iT - 256T^{2}
3 1−84.5T+6.56e3T2 1 - 84.5T + 6.56e3T^{2}
5 1−63.0T+3.90e5T2 1 - 63.0T + 3.90e5T^{2}
7 1−1.66e3iT−5.76e6T2 1 - 1.66e3iT - 5.76e6T^{2}
13 1+1.61e4iT−8.15e8T2 1 + 1.61e4iT - 8.15e8T^{2}
17 1+5.82e4iT−6.97e9T2 1 + 5.82e4iT - 6.97e9T^{2}
19 1+1.72e4iT−1.69e10T2 1 + 1.72e4iT - 1.69e10T^{2}
23 1−2.90e5T+7.83e10T2 1 - 2.90e5T + 7.83e10T^{2}
29 1+1.15e6iT−5.00e11T2 1 + 1.15e6iT - 5.00e11T^{2}
31 1+5.04e5T+8.52e11T2 1 + 5.04e5T + 8.52e11T^{2}
37 1+5.62e5T+3.51e12T2 1 + 5.62e5T + 3.51e12T^{2}
41 1−9.57e5iT−7.98e12T2 1 - 9.57e5iT - 7.98e12T^{2}
43 1−6.28e6iT−1.16e13T2 1 - 6.28e6iT - 1.16e13T^{2}
47 1−7.14e6T+2.38e13T2 1 - 7.14e6T + 2.38e13T^{2}
53 1+5.44e6T+6.22e13T2 1 + 5.44e6T + 6.22e13T^{2}
59 1+1.63e7T+1.46e14T2 1 + 1.63e7T + 1.46e14T^{2}
61 1−1.66e7iT−1.91e14T2 1 - 1.66e7iT - 1.91e14T^{2}
67 1−2.30e7T+4.06e14T2 1 - 2.30e7T + 4.06e14T^{2}
71 1+1.66e7T+6.45e14T2 1 + 1.66e7T + 6.45e14T^{2}
73 1−5.01e7iT−8.06e14T2 1 - 5.01e7iT - 8.06e14T^{2}
79 1+4.05e7iT−1.51e15T2 1 + 4.05e7iT - 1.51e15T^{2}
83 1+4.65e7iT−2.25e15T2 1 + 4.65e7iT - 2.25e15T^{2}
89 1−9.49e7T+3.93e15T2 1 - 9.49e7T + 3.93e15T^{2}
97 1+3.42e7T+7.83e15T2 1 + 3.42e7T + 7.83e15T^{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

−19.00253606306714551156566277056, −17.27645964664228780005052606454, −15.80070137687546990254844183104, −14.81267030084517479038022361942, −13.63746380045899782518112603201, −11.47056105360012306615555929408, −9.053650980498135914826333118106, −7.82519504149597608241120338744, −5.85417590102816912254671156201, −2.72988702990547371793035530900, 1.86310637002698378178010381099, 3.69023494888069122645372678250, 7.18077094961160684637032025729, 9.236785279790319063992905918690, 10.71780601541548590953311467667, 12.41707877893598020109933809826, 13.90374732011962460092528261892, 15.25423954456883422903737431540, 17.05078024156280932306721358951, 18.97730020908900731616768106635

Graph of the ZZ-function along the critical line