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Properties

Label 2-350-1.1-c1-0-0
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 0.53200103330.5320010333
L(12)L(\frac12) ≈\approx 0.53200103330.5320010333
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

−11.29554923076180718025028882069, −10.55325367837767017951933665498, −10.01439033783112463655606599604, −8.664179296918683319137492382330, −7.70971377206101767237854140637, −6.53697423851433080849603310003, −5.83104606741587686350813742208, −4.80557649010691109432273744906, −2.97562386654069150960632236205, −0.862503666228635144429925468519, 0.862503666228635144429925468519, 2.97562386654069150960632236205, 4.80557649010691109432273744906, 5.83104606741587686350813742208, 6.53697423851433080849603310003, 7.70971377206101767237854140637, 8.664179296918683319137492382330, 10.01439033783112463655606599604, 10.55325367837767017951933665498, 11.29554923076180718025028882069

Graph of the ZZ-function along the critical line