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Properties

Label 2-1911-13.12-c1-0-0
Degree 22
Conductor 19111911
Sign 0.410+0.911i0.410 + 0.911i
Analytic cond. 15.259415.2594
Root an. cond. 3.906323.90632
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  + 1.48i·2-s − 3-s − 0.193·4-s + 4.15i·5-s − 1.48i·6-s + 2.67i·8-s + 9-s − 6.15·10-s − 3.19i·11-s + 0.193·12-s + (−1.48 − 3.28i)13-s − 4.15i·15-s − 4.35·16-s − 3.35·17-s + 1.48i·18-s − 2.38i·19-s + ⋯
L(s)  = 1  + 1.04i·2-s − 0.577·3-s − 0.0969·4-s + 1.85i·5-s − 0.604i·6-s + 0.945i·8-s + 0.333·9-s − 1.94·10-s − 0.963i·11-s + 0.0559·12-s + (−0.410 − 0.911i)13-s − 1.07i·15-s − 1.08·16-s − 0.812·17-s + 0.349i·18-s − 0.547i·19-s + ⋯

Functional equation

Λ(s)=(1911s/2 ΓC(s) L(s)=((0.410+0.911i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.410 + 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1911s/2 ΓC(s+1/2) L(s)=((0.410+0.911i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.410 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 19111911    =    3⋅72⋅133 \cdot 7^{2} \cdot 13
Sign: 0.410+0.911i0.410 + 0.911i
Analytic conductor: 15.259415.2594
Root analytic conductor: 3.906323.90632
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1911(883,⋅)\chi_{1911} (883, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1911, ( :1/2), 0.410+0.911i)(2,\ 1911,\ (\ :1/2),\ 0.410 + 0.911i)

Particular Values

L(1)L(1) ≈\approx 0.13831947520.1383194752
L(12)L(\frac12) ≈\approx 0.13831947520.1383194752
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)
bad3 1+T 1 + T
7 1 1
13 1+(1.48+3.28i)T 1 + (1.48 + 3.28i)T
good2 1−1.48iT−2T2 1 - 1.48iT - 2T^{2}
5 1−4.15iT−5T2 1 - 4.15iT - 5T^{2}
11 1+3.19iT−11T2 1 + 3.19iT - 11T^{2}
17 1+3.35T+17T2 1 + 3.35T + 17T^{2}
19 1+2.38iT−19T2 1 + 2.38iT - 19T^{2}
23 1−0.387T+23T2 1 - 0.387T + 23T^{2}
29 1+7.92T+29T2 1 + 7.92T + 29T^{2}
31 1+10.7iT−31T2 1 + 10.7iT - 31T^{2}
37 1+1.61iT−37T2 1 + 1.61iT - 37T^{2}
41 1+1.45iT−41T2 1 + 1.45iT - 41T^{2}
43 1−1.92T+43T2 1 - 1.92T + 43T^{2}
47 1−3.76iT−47T2 1 - 3.76iT - 47T^{2}
53 1+6T+53T2 1 + 6T + 53T^{2}
59 1−6.15iT−59T2 1 - 6.15iT - 59T^{2}
61 1+14.4T+61T2 1 + 14.4T + 61T^{2}
67 1−5.61iT−67T2 1 - 5.61iT - 67T^{2}
71 1−11.8iT−71T2 1 - 11.8iT - 71T^{2}
73 1−15.6iT−73T2 1 - 15.6iT - 73T^{2}
79 1+8.96T+79T2 1 + 8.96T + 79T^{2}
83 1+6.99iT−83T2 1 + 6.99iT - 83T^{2}
89 1+0.932iT−89T2 1 + 0.932iT - 89T^{2}
97 1−3.35iT−97T2 1 - 3.35iT - 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

−9.941918735166031664149252935758, −8.924906288237989465269232271030, −7.76390828717565526411808748743, −7.42373397788222276127540892779, −6.66678857614951176514905661678, −5.94880250836217116333678227625, −5.59903555944447002956213963203, −4.21289177338175511976256952723, −3.03213426754001186191992816323, −2.29290838490209376026516058917, 0.05006247155408423926986687019, 1.52017363996875493607568444759, 1.91661352727798043243506738908, 3.59481036421459030761546156561, 4.61065575765795872003783602654, 4.84735414181332234720780930689, 6.07359402085860734872736703763, 6.98219337473295574473882349987, 7.83479066128507489051001126712, 9.087624368439693920090233367051

Graph of the ZZ-function along the critical line