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Properties

Label 2-224-224.69-c0-0-0
Degree 22
Conductor 224224
Sign 0.555−0.831i0.555 - 0.831i
Analytic cond. 0.1117900.111790
Root an. cond. 0.3343500.334350
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 0.707i)2-s + 1.00i·4-s + (−0.707 − 0.707i)7-s + (−0.707 + 0.707i)8-s + (−0.707 + 0.707i)9-s + (0.707 − 1.70i)11-s − 1.00i·14-s − 1.00·16-s − 1.00·18-s + (1.70 − 0.707i)22-s + (−1 + i)23-s + (0.707 + 0.707i)25-s + (0.707 − 0.707i)28-s + (−0.707 − 1.70i)29-s + (−0.707 − 0.707i)32-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + 1.00i·4-s + (−0.707 − 0.707i)7-s + (−0.707 + 0.707i)8-s + (−0.707 + 0.707i)9-s + (0.707 − 1.70i)11-s − 1.00i·14-s − 1.00·16-s − 1.00·18-s + (1.70 − 0.707i)22-s + (−1 + i)23-s + (0.707 + 0.707i)25-s + (0.707 − 0.707i)28-s + (−0.707 − 1.70i)29-s + (−0.707 − 0.707i)32-s + ⋯

Functional equation

Λ(s)=(224s/2 ΓC(s) L(s)=((0.555−0.831i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(224s/2 ΓC(s) L(s)=((0.555−0.831i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 224224    =    25⋅72^{5} \cdot 7
Sign: 0.555−0.831i0.555 - 0.831i
Analytic conductor: 0.1117900.111790
Root analytic conductor: 0.3343500.334350
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ224(69,⋅)\chi_{224} (69, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 224, ( :0), 0.555−0.831i)(2,\ 224,\ (\ :0),\ 0.555 - 0.831i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.91478789000.9147878900
L(12)L(\frac12) ≈\approx 0.91478789000.9147878900
L(1)L(1) 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+(−0.707−0.707i)T 1 + (-0.707 - 0.707i)T
7 1+(0.707+0.707i)T 1 + (0.707 + 0.707i)T
good3 1+(0.707−0.707i)T2 1 + (0.707 - 0.707i)T^{2}
5 1+(−0.707−0.707i)T2 1 + (-0.707 - 0.707i)T^{2}
11 1+(−0.707+1.70i)T+(−0.707−0.707i)T2 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2}
13 1+(−0.707+0.707i)T2 1 + (-0.707 + 0.707i)T^{2}
17 1+T2 1 + T^{2}
19 1+(−0.707+0.707i)T2 1 + (-0.707 + 0.707i)T^{2}
23 1+(1−i)T−iT2 1 + (1 - i)T - iT^{2}
29 1+(0.707+1.70i)T+(−0.707+0.707i)T2 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2}
31 1−T2 1 - T^{2}
37 1+(−0.707−0.292i)T+(0.707+0.707i)T2 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2}
41 1+iT2 1 + iT^{2}
43 1+(0.292−0.707i)T+(−0.707−0.707i)T2 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2}
47 1+T2 1 + T^{2}
53 1+(0.292−0.707i)T+(−0.707−0.707i)T2 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2}
59 1+(−0.707−0.707i)T2 1 + (-0.707 - 0.707i)T^{2}
61 1+(0.707−0.707i)T2 1 + (0.707 - 0.707i)T^{2}
67 1+(−0.292−0.707i)T+(−0.707+0.707i)T2 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2}
71 1+iT2 1 + iT^{2}
73 1+iT2 1 + iT^{2}
79 1−1.41iT−T2 1 - 1.41iT - T^{2}
83 1+(−0.707+0.707i)T2 1 + (-0.707 + 0.707i)T^{2}
89 1−iT2 1 - iT^{2}
97 1−T2 1 - 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

−13.01896047236179747630980474753, −11.62413417223418457936967062638, −11.08933817524096374040867117362, −9.558664557522519020388867669763, −8.434536957138598386964682300441, −7.58349170248394100224433670765, −6.29893581866913974948321944843, −5.63213365356674983441617051222, −4.05555792793186239881001508243, −3.05489877579825797630944806539, 2.21337394587585197564071831515, 3.55072378156165014876097734764, 4.81090955923048536674171304725, 6.10004080784686032476692554731, 6.88124373243444318155137645877, 8.823579721935180019121213818173, 9.545982489762076035360758359071, 10.44510991769651225499325994251, 11.74811795196556075540511044261, 12.35448965942733587960770380009

Graph of the ZZ-function along the critical line