↑

Properties

Label 2-950-1.1-c1-0-7
Degree 22
Conductor 950950
Sign 11
Analytic cond. 7.585787.58578
Root an. cond. 2.754232.75423
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 0.414·3-s + 4-s + 0.414·6-s + 4.41·7-s − 8-s − 2.82·9-s − 1.41·11-s − 0.414·12-s + 5.82·13-s − 4.41·14-s + 16-s + 17-s + 2.82·18-s − 19-s − 1.82·21-s + 1.41·22-s + 0.757·23-s + 0.414·24-s − 5.82·26-s + 2.41·27-s + 4.41·28-s − 0.171·29-s + 6.24·31-s − 32-s + 0.585·33-s − 34-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.239·3-s + 0.5·4-s + 0.169·6-s + 1.66·7-s − 0.353·8-s − 0.942·9-s − 0.426·11-s − 0.119·12-s + 1.61·13-s − 1.17·14-s + 0.250·16-s + 0.242·17-s + 0.666·18-s − 0.229·19-s − 0.398·21-s + 0.301·22-s + 0.157·23-s + 0.0845·24-s − 1.14·26-s + 0.464·27-s + 0.834·28-s − 0.0318·29-s + 1.12·31-s − 0.176·32-s + 0.101·33-s − 0.171·34-s + ⋯

Functional equation

Λ(s)=(950s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(950s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 950950    =    2⋅52⋅192 \cdot 5^{2} \cdot 19
Sign: 11
Analytic conductor: 7.585787.58578
Root analytic conductor: 2.754232.75423
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 950, ( :1/2), 1)(2,\ 950,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.2391602501.239160250
L(12)L(\frac12) ≈\approx 1.2391602501.239160250
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
19 1+T 1 + T
good3 1+0.414T+3T2 1 + 0.414T + 3T^{2}
7 1−4.41T+7T2 1 - 4.41T + 7T^{2}
11 1+1.41T+11T2 1 + 1.41T + 11T^{2}
13 1−5.82T+13T2 1 - 5.82T + 13T^{2}
17 1−T+17T2 1 - T + 17T^{2}
23 1−0.757T+23T2 1 - 0.757T + 23T^{2}
29 1+0.171T+29T2 1 + 0.171T + 29T^{2}
31 1−6.24T+31T2 1 - 6.24T + 31T^{2}
37 1+8.48T+37T2 1 + 8.48T + 37T^{2}
41 1+4.24T+41T2 1 + 4.24T + 41T^{2}
43 1−1.75T+43T2 1 - 1.75T + 43T^{2}
47 1+47T2 1 + 47T^{2}
53 1−5.48T+53T2 1 - 5.48T + 53T^{2}
59 1−6.89T+59T2 1 - 6.89T + 59T^{2}
61 1−14.2T+61T2 1 - 14.2T + 61T^{2}
67 1−4.75T+67T2 1 - 4.75T + 67T^{2}
71 1+13.4T+71T2 1 + 13.4T + 71T^{2}
73 1−11.4T+73T2 1 - 11.4T + 73T^{2}
79 1+6.48T+79T2 1 + 6.48T + 79T^{2}
83 1−14.4T+83T2 1 - 14.4T + 83T^{2}
89 1−7.07T+89T2 1 - 7.07T + 89T^{2}
97 1+0.343T+97T2 1 + 0.343T + 97T^{2}
show more
show less
   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

−10.28077010141182019098845748019, −8.831862100813789181510536703591, −8.458968035015643325998359679496, −7.85092908730108193919178449370, −6.69205257398238111475743537251, −5.70804418276667345505510457016, −4.97743752742474768646814718854, −3.63150975539577213875955977149, −2.24874253997621326318976801540, −1.04897719567186889567602968690, 1.04897719567186889567602968690, 2.24874253997621326318976801540, 3.63150975539577213875955977149, 4.97743752742474768646814718854, 5.70804418276667345505510457016, 6.69205257398238111475743537251, 7.85092908730108193919178449370, 8.458968035015643325998359679496, 8.831862100813789181510536703591, 10.28077010141182019098845748019

Graph of the ZZ-function along the critical line