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Properties

Label 2-350-1.1-c1-0-8
Degree 22
Conductor 350350
Sign 11
Analytic cond. 2.794762.79476
Root an. cond. 1.671751.67175
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2.44·3-s + 4-s + 2.44·6-s + 7-s + 8-s + 2.99·9-s − 4.89·11-s + 2.44·12-s − 4.44·13-s + 14-s + 16-s − 2·17-s + 2.99·18-s + 1.55·19-s + 2.44·21-s − 4.89·22-s − 2.89·23-s + 2.44·24-s − 4.44·26-s + 28-s + 6.89·29-s + 8.89·31-s + 32-s − 11.9·33-s − 2·34-s + 2.99·36-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.41·3-s + 0.5·4-s + 0.999·6-s + 0.377·7-s + 0.353·8-s + 0.999·9-s − 1.47·11-s + 0.707·12-s − 1.23·13-s + 0.267·14-s + 0.250·16-s − 0.485·17-s + 0.707·18-s + 0.355·19-s + 0.534·21-s − 1.04·22-s − 0.604·23-s + 0.499·24-s − 0.872·26-s + 0.188·28-s + 1.28·29-s + 1.59·31-s + 0.176·32-s − 2.08·33-s − 0.342·34-s + 0.499·36-s + ⋯

Functional equation

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

Invariants

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

Particular Values

L(1)L(1) ≈\approx 2.8309825302.830982530
L(12)L(\frac12) ≈\approx 2.8309825302.830982530
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−T 1 - T
5 1 1
7 1−T 1 - T
good3 1−2.44T+3T2 1 - 2.44T + 3T^{2}
11 1+4.89T+11T2 1 + 4.89T + 11T^{2}
13 1+4.44T+13T2 1 + 4.44T + 13T^{2}
17 1+2T+17T2 1 + 2T + 17T^{2}
19 1−1.55T+19T2 1 - 1.55T + 19T^{2}
23 1+2.89T+23T2 1 + 2.89T + 23T^{2}
29 1−6.89T+29T2 1 - 6.89T + 29T^{2}
31 1−8.89T+31T2 1 - 8.89T + 31T^{2}
37 1+2T+37T2 1 + 2T + 37T^{2}
41 1+1.10T+41T2 1 + 1.10T + 41T^{2}
43 1−0.898T+43T2 1 - 0.898T + 43T^{2}
47 1+8.89T+47T2 1 + 8.89T + 47T^{2}
53 1−10.8T+53T2 1 - 10.8T + 53T^{2}
59 1+1.55T+59T2 1 + 1.55T + 59T^{2}
61 1−3.55T+61T2 1 - 3.55T + 61T^{2}
67 1−8T+67T2 1 - 8T + 67T^{2}
71 1+1.10T+71T2 1 + 1.10T + 71T^{2}
73 1+2.89T+73T2 1 + 2.89T + 73T^{2}
79 1−6.89T+79T2 1 - 6.89T + 79T^{2}
83 1−2.44T+83T2 1 - 2.44T + 83T^{2}
89 1+10T+89T2 1 + 10T + 89T^{2}
97 1+15.7T+97T2 1 + 15.7T + 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

−11.70456166423393402195005342402, −10.40695359032285207903865428901, −9.738844466819335119483763061725, −8.375924916232554564968780107376, −7.88805460590994197147068727888, −6.86512930623403544071985121749, −5.30743194698470000684988949513, −4.39664725359770703549679315738, −2.95437356862933556349589288599, −2.28923518601867728793453841962, 2.28923518601867728793453841962, 2.95437356862933556349589288599, 4.39664725359770703549679315738, 5.30743194698470000684988949513, 6.86512930623403544071985121749, 7.88805460590994197147068727888, 8.375924916232554564968780107376, 9.738844466819335119483763061725, 10.40695359032285207903865428901, 11.70456166423393402195005342402

Graph of the ZZ-function along the critical line