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Properties

Label 2-350-5.4-c1-0-0
Degree 22
Conductor 350350
Sign −0.894+0.447i-0.894 + 0.447i
Analytic cond. 2.794762.79476
Root an. cond. 1.671751.67175
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  + i·2-s + 3i·3-s − 4-s − 3·6-s − i·7-s − i·8-s − 6·9-s − 5·11-s − 3i·12-s + 6i·13-s + 14-s + 16-s − i·17-s − 6i·18-s + 3·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.73i·3-s − 0.5·4-s − 1.22·6-s − 0.377i·7-s − 0.353i·8-s − 2·9-s − 1.50·11-s − 0.866i·12-s + 1.66i·13-s + 0.267·14-s + 0.250·16-s − 0.242i·17-s − 1.41i·18-s + 0.688·19-s + ⋯

Functional equation

Λ(s)=(350s/2 ΓC(s) L(s)=((−0.894+0.447i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(350s/2 ΓC(s+1/2) L(s)=((−0.894+0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 350350    =    2⋅52⋅72 \cdot 5^{2} \cdot 7
Sign: −0.894+0.447i-0.894 + 0.447i
Analytic conductor: 2.794762.79476
Root analytic conductor: 1.671751.67175
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ350(99,⋅)\chi_{350} (99, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 350, ( :1/2), −0.894+0.447i)(2,\ 350,\ (\ :1/2),\ -0.894 + 0.447i)

Particular Values

L(1)L(1) ≈\approx 0.210675−0.892434i0.210675 - 0.892434i
L(12)L(\frac12) ≈\approx 0.210675−0.892434i0.210675 - 0.892434i
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−iT 1 - iT
5 1 1
7 1+iT 1 + iT
good3 1−3iT−3T2 1 - 3iT - 3T^{2}
11 1+5T+11T2 1 + 5T + 11T^{2}
13 1−6iT−13T2 1 - 6iT - 13T^{2}
17 1+iT−17T2 1 + iT - 17T^{2}
19 1−3T+19T2 1 - 3T + 19T^{2}
23 1−23T2 1 - 23T^{2}
29 1−6T+29T2 1 - 6T + 29T^{2}
31 1+4T+31T2 1 + 4T + 31T^{2}
37 1−8iT−37T2 1 - 8iT - 37T^{2}
41 1−11T+41T2 1 - 11T + 41T^{2}
43 1−8iT−43T2 1 - 8iT - 43T^{2}
47 1−2iT−47T2 1 - 2iT - 47T^{2}
53 1+4iT−53T2 1 + 4iT - 53T^{2}
59 1+4T+59T2 1 + 4T + 59T^{2}
61 1+2T+61T2 1 + 2T + 61T^{2}
67 1−9iT−67T2 1 - 9iT - 67T^{2}
71 1+10T+71T2 1 + 10T + 71T^{2}
73 1−7iT−73T2 1 - 7iT - 73T^{2}
79 1−2T+79T2 1 - 2T + 79T^{2}
83 1+11iT−83T2 1 + 11iT - 83T^{2}
89 1−11T+89T2 1 - 11T + 89T^{2}
97 1+10iT−97T2 1 + 10iT - 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

−11.71213532525521739404529331675, −10.85847408081630898113138468309, −9.994343734392532640515735584001, −9.370667495287345675425294175021, −8.432901335985605660532466935114, −7.34160332609263357086643054592, −6.00923260737645566883775136857, −4.89340268853846218069727323280, −4.35538657252140008829443741148, −3.01173303664762003186045602547, 0.62531841110652152098686894891, 2.25592963732092920140091820731, 3.09017020285620483598189848717, 5.26795596119065845092726418093, 5.95850393584290940415092698715, 7.51444651436689669043190127183, 7.900121616147704622314811869964, 8.921340809713994409888151596341, 10.35013153478523174428802262470, 11.03248914126200209180492012179

Graph of the ZZ-function along the critical line