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Properties

Label 2-684-19.16-c1-0-1
Degree 22
Conductor 684684
Sign −0.575−0.817i-0.575 - 0.817i
Analytic cond. 5.461765.46176
Root an. cond. 2.337042.33704
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  + (−0.485 + 2.75i)5-s + (1.68 + 2.91i)7-s + (−0.258 + 0.447i)11-s + (−4.37 − 1.59i)13-s + (0.735 + 0.617i)17-s + (−3.12 + 3.04i)19-s + (−0.629 − 3.57i)23-s + (−2.63 − 0.958i)25-s + (−6.21 + 5.21i)29-s + (−2.38 − 4.13i)31-s + (−8.83 + 3.21i)35-s + 9.13·37-s + (6.54 − 2.38i)41-s + (−0.817 + 4.63i)43-s + (−10.4 + 8.74i)47-s + ⋯
L(s)  = 1  + (−0.216 + 1.23i)5-s + (0.636 + 1.10i)7-s + (−0.0778 + 0.134i)11-s + (−1.21 − 0.441i)13-s + (0.178 + 0.149i)17-s + (−0.715 + 0.698i)19-s + (−0.131 − 0.744i)23-s + (−0.526 − 0.191i)25-s + (−1.15 + 0.968i)29-s + (−0.429 − 0.743i)31-s + (−1.49 + 0.543i)35-s + 1.50·37-s + (1.02 − 0.372i)41-s + (−0.124 + 0.706i)43-s + (−1.51 + 1.27i)47-s + ⋯

Functional equation

Λ(s)=(684s/2 ΓC(s) L(s)=((−0.575−0.817i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.575 - 0.817i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(684s/2 ΓC(s+1/2) L(s)=((−0.575−0.817i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.575 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 684684    =    22⋅32⋅192^{2} \cdot 3^{2} \cdot 19
Sign: −0.575−0.817i-0.575 - 0.817i
Analytic conductor: 5.461765.46176
Root analytic conductor: 2.337042.33704
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ684(73,⋅)\chi_{684} (73, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 684, ( :1/2), −0.575−0.817i)(2,\ 684,\ (\ :1/2),\ -0.575 - 0.817i)

Particular Values

L(1)L(1) ≈\approx 0.530484+1.02249i0.530484 + 1.02249i
L(12)L(\frac12) ≈\approx 0.530484+1.02249i0.530484 + 1.02249i
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
3 1 1
19 1+(3.12−3.04i)T 1 + (3.12 - 3.04i)T
good5 1+(0.485−2.75i)T+(−4.69−1.71i)T2 1 + (0.485 - 2.75i)T + (-4.69 - 1.71i)T^{2}
7 1+(−1.68−2.91i)T+(−3.5+6.06i)T2 1 + (-1.68 - 2.91i)T + (-3.5 + 6.06i)T^{2}
11 1+(0.258−0.447i)T+(−5.5−9.52i)T2 1 + (0.258 - 0.447i)T + (-5.5 - 9.52i)T^{2}
13 1+(4.37+1.59i)T+(9.95+8.35i)T2 1 + (4.37 + 1.59i)T + (9.95 + 8.35i)T^{2}
17 1+(−0.735−0.617i)T+(2.95+16.7i)T2 1 + (-0.735 - 0.617i)T + (2.95 + 16.7i)T^{2}
23 1+(0.629+3.57i)T+(−21.6+7.86i)T2 1 + (0.629 + 3.57i)T + (-21.6 + 7.86i)T^{2}
29 1+(6.21−5.21i)T+(5.03−28.5i)T2 1 + (6.21 - 5.21i)T + (5.03 - 28.5i)T^{2}
31 1+(2.38+4.13i)T+(−15.5+26.8i)T2 1 + (2.38 + 4.13i)T + (-15.5 + 26.8i)T^{2}
37 1−9.13T+37T2 1 - 9.13T + 37T^{2}
41 1+(−6.54+2.38i)T+(31.4−26.3i)T2 1 + (-6.54 + 2.38i)T + (31.4 - 26.3i)T^{2}
43 1+(0.817−4.63i)T+(−40.4−14.7i)T2 1 + (0.817 - 4.63i)T + (-40.4 - 14.7i)T^{2}
47 1+(10.4−8.74i)T+(8.16−46.2i)T2 1 + (10.4 - 8.74i)T + (8.16 - 46.2i)T^{2}
53 1+(−1.20−6.81i)T+(−49.8+18.1i)T2 1 + (-1.20 - 6.81i)T + (-49.8 + 18.1i)T^{2}
59 1+(−10.9−9.19i)T+(10.2+58.1i)T2 1 + (-10.9 - 9.19i)T + (10.2 + 58.1i)T^{2}
61 1+(−1.05−6.00i)T+(−57.3+20.8i)T2 1 + (-1.05 - 6.00i)T + (-57.3 + 20.8i)T^{2}
67 1+(−3.38+2.84i)T+(11.6−65.9i)T2 1 + (-3.38 + 2.84i)T + (11.6 - 65.9i)T^{2}
71 1+(−1.66+9.42i)T+(−66.7−24.2i)T2 1 + (-1.66 + 9.42i)T + (-66.7 - 24.2i)T^{2}
73 1+(−5.65+2.05i)T+(55.9−46.9i)T2 1 + (-5.65 + 2.05i)T + (55.9 - 46.9i)T^{2}
79 1+(9.85−3.58i)T+(60.5−50.7i)T2 1 + (9.85 - 3.58i)T + (60.5 - 50.7i)T^{2}
83 1+(2.39+4.14i)T+(−41.5+71.8i)T2 1 + (2.39 + 4.14i)T + (-41.5 + 71.8i)T^{2}
89 1+(−8.30−3.02i)T+(68.1+57.2i)T2 1 + (-8.30 - 3.02i)T + (68.1 + 57.2i)T^{2}
97 1+(−8.46−7.10i)T+(16.8+95.5i)T2 1 + (-8.46 - 7.10i)T + (16.8 + 95.5i)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

−10.85423641613247017534713117393, −10.00498439199056605839717631184, −9.103375513901357614364252762752, −7.992418922331131979312841369803, −7.42359831665845473045992885509, −6.29590065389037938142067511809, −5.48017719731709420764591624585, −4.31892153844660135924713506355, −2.93501310056892336600863798177, −2.15015519457790365016376344980, 0.59366379455735406325635346373, 2.03760663253561296552494120745, 3.85819895284241338656695765235, 4.66150209408773096195092201645, 5.34756455719911461518697971375, 6.83190163989672656962340099438, 7.65757062684222607501100996447, 8.362898220685349861918309222220, 9.373023991550634610995992379310, 10.03251183531345553209498852576

Graph of the ZZ-function along the critical line