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Properties

Label 2-2156-77.10-c1-0-5
Degree 22
Conductor 21562156
Sign 0.426−0.904i0.426 - 0.904i
Analytic cond. 17.215717.2157
Root an. cond. 4.149184.14918
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  + (−1.60 − 0.923i)3-s + (1.60 − 0.923i)5-s + (0.207 + 0.358i)9-s + (−2.23 − 2.44i)11-s − 5.84·13-s − 3.41·15-s + (−1.21 + 2.09i)17-s + (4.13 + 7.15i)19-s + (−2.12 − 3.67i)23-s + (−0.792 + 1.37i)25-s + 4.77i·27-s − 4.47i·29-s + (7.06 + 4.07i)31-s + (1.32 + 5.98i)33-s + (3.41 + 5.91i)37-s + ⋯
L(s)  = 1  + (−0.923 − 0.533i)3-s + (0.715 − 0.413i)5-s + (0.0690 + 0.119i)9-s + (−0.674 − 0.737i)11-s − 1.62·13-s − 0.881·15-s + (−0.293 + 0.508i)17-s + (0.947 + 1.64i)19-s + (−0.442 − 0.766i)23-s + (−0.158 + 0.274i)25-s + 0.919i·27-s − 0.830i·29-s + (1.26 + 0.732i)31-s + (0.230 + 1.04i)33-s + (0.561 + 0.972i)37-s + ⋯

Functional equation

Λ(s)=(2156s/2 ΓC(s) L(s)=((0.426−0.904i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.426 - 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2156s/2 ΓC(s+1/2) L(s)=((0.426−0.904i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.426 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21562156    =    22⋅72⋅112^{2} \cdot 7^{2} \cdot 11
Sign: 0.426−0.904i0.426 - 0.904i
Analytic conductor: 17.215717.2157
Root analytic conductor: 4.149184.14918
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2156(2089,⋅)\chi_{2156} (2089, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2156, ( :1/2), 0.426−0.904i)(2,\ 2156,\ (\ :1/2),\ 0.426 - 0.904i)

Particular Values

L(1)L(1) ≈\approx 0.60308378910.6030837891
L(12)L(\frac12) ≈\approx 0.60308378910.6030837891
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
7 1 1
11 1+(2.23+2.44i)T 1 + (2.23 + 2.44i)T
good3 1+(1.60+0.923i)T+(1.5+2.59i)T2 1 + (1.60 + 0.923i)T + (1.5 + 2.59i)T^{2}
5 1+(−1.60+0.923i)T+(2.5−4.33i)T2 1 + (-1.60 + 0.923i)T + (2.5 - 4.33i)T^{2}
13 1+5.84T+13T2 1 + 5.84T + 13T^{2}
17 1+(1.21−2.09i)T+(−8.5−14.7i)T2 1 + (1.21 - 2.09i)T + (-8.5 - 14.7i)T^{2}
19 1+(−4.13−7.15i)T+(−9.5+16.4i)T2 1 + (-4.13 - 7.15i)T + (-9.5 + 16.4i)T^{2}
23 1+(2.12+3.67i)T+(−11.5+19.9i)T2 1 + (2.12 + 3.67i)T + (-11.5 + 19.9i)T^{2}
29 1+4.47iT−29T2 1 + 4.47iT - 29T^{2}
31 1+(−7.06−4.07i)T+(15.5+26.8i)T2 1 + (-7.06 - 4.07i)T + (15.5 + 26.8i)T^{2}
37 1+(−3.41−5.91i)T+(−18.5+32.0i)T2 1 + (-3.41 - 5.91i)T + (-18.5 + 32.0i)T^{2}
41 1+9.26T+41T2 1 + 9.26T + 41T^{2}
43 1−1.85iT−43T2 1 - 1.85iT - 43T^{2}
47 1+(−4.25+2.45i)T+(23.5−40.7i)T2 1 + (-4.25 + 2.45i)T + (23.5 - 40.7i)T^{2}
53 1+(3.53−6.12i)T+(−26.5−45.8i)T2 1 + (3.53 - 6.12i)T + (-26.5 - 45.8i)T^{2}
59 1+(3.47+2.00i)T+(29.5+51.0i)T2 1 + (3.47 + 2.00i)T + (29.5 + 51.0i)T^{2}
61 1+(−1.21−2.09i)T+(−30.5+52.8i)T2 1 + (-1.21 - 2.09i)T + (-30.5 + 52.8i)T^{2}
67 1+(−5.53+9.58i)T+(−33.5−58.0i)T2 1 + (-5.53 + 9.58i)T + (-33.5 - 58.0i)T^{2}
71 1−3.75T+71T2 1 - 3.75T + 71T^{2}
73 1+(5.34−9.25i)T+(−36.5−63.2i)T2 1 + (5.34 - 9.25i)T + (-36.5 - 63.2i)T^{2}
79 1+(−11.6+6.70i)T+(39.5−68.4i)T2 1 + (-11.6 + 6.70i)T + (39.5 - 68.4i)T^{2}
83 1−4.84T+83T2 1 - 4.84T + 83T^{2}
89 1+(−3.86+2.23i)T+(44.5−77.0i)T2 1 + (-3.86 + 2.23i)T + (44.5 - 77.0i)T^{2}
97 1+5.35iT−97T2 1 + 5.35iT - 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

−9.361498898301545447962337982968, −8.252398697884258963799464387671, −7.72873041158819036350919047556, −6.64010992800712577694081013164, −6.03688487822513837377258108883, −5.37871820000042513984114971986, −4.72433886217457433822222105899, −3.32614977666911909735788117357, −2.19852240790898094355514764745, −1.06409868353984020759810844362, 0.26120230318499306612644704569, 2.21470322723820414844673784289, 2.81379959274154704848980542700, 4.41473470304177018579762903901, 5.08153327803172700149031915784, 5.48524377564650213493068631194, 6.61607579635947561099650084619, 7.22121908468530676916105365202, 8.011385737100841185227400994482, 9.422287469483145431767563811540

Graph of the ZZ-function along the critical line