↑

Properties

Label 2-303450-1.1-c1-0-62
Degree 22
Conductor 303450303450
Sign 11
Analytic cond. 2423.062423.06
Root an. cond. 49.224549.2245
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s + 4·11-s − 12-s + 5·13-s + 14-s + 16-s − 18-s + 2·19-s + 21-s − 4·22-s − 3·23-s + 24-s − 5·26-s − 27-s − 28-s + 8·29-s + 6·31-s − 32-s − 4·33-s + 36-s + 37-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 1.20·11-s − 0.288·12-s + 1.38·13-s + 0.267·14-s + 1/4·16-s − 0.235·18-s + 0.458·19-s + 0.218·21-s − 0.852·22-s − 0.625·23-s + 0.204·24-s − 0.980·26-s − 0.192·27-s − 0.188·28-s + 1.48·29-s + 1.07·31-s − 0.176·32-s − 0.696·33-s + 1/6·36-s + 0.164·37-s + ⋯

Functional equation

Λ(s)=(303450s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(303450s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 303450303450    =    2⋅3⋅52⋅7⋅1722 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 17^{2}
Sign: 11
Analytic conductor: 2423.062423.06
Root analytic conductor: 49.224549.2245
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 303450, ( :1/2), 1)(2,\ 303450,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.1990291642.199029164
L(12)L(\frac12) ≈\approx 2.1990291642.199029164
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
3 1+T 1 + T
5 1 1
7 1+T 1 + T
17 1 1
good11 1−4T+pT2 1 - 4 T + p T^{2}
13 1−5T+pT2 1 - 5 T + p T^{2}
19 1−2T+pT2 1 - 2 T + p T^{2}
23 1+3T+pT2 1 + 3 T + p T^{2}
29 1−8T+pT2 1 - 8 T + p T^{2}
31 1−6T+pT2 1 - 6 T + p T^{2}
37 1−T+pT2 1 - T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1−7T+pT2 1 - 7 T + p T^{2}
53 1+8T+pT2 1 + 8 T + p T^{2}
59 1−5T+pT2 1 - 5 T + p T^{2}
61 1−5T+pT2 1 - 5 T + p T^{2}
67 1+16T+pT2 1 + 16 T + p T^{2}
71 1+3T+pT2 1 + 3 T + p T^{2}
73 1+3T+pT2 1 + 3 T + p T^{2}
79 1−10T+pT2 1 - 10 T + p T^{2}
83 1+3T+pT2 1 + 3 T + p T^{2}
89 1−14T+pT2 1 - 14 T + p T^{2}
97 1+7T+pT2 1 + 7 T + p T^{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

−12.50487983397450, −12.03528230772609, −11.75350553436348, −11.31998030541539, −10.81481033940600, −10.32298375026147, −10.05478315553848, −9.370764214645245, −9.051369984613982, −8.634039074716019, −8.074891999306370, −7.599976216244607, −7.039190904476509, −6.459440907629924, −6.152908063276028, −5.967653352318863, −5.151006590442446, −4.454027452905297, −4.053665426600552, −3.499639524681135, −2.902411775371101, −2.268589858801701, −1.419056196189427, −1.052878118729228, −0.5562752828040814, 0.5562752828040814, 1.052878118729228, 1.419056196189427, 2.268589858801701, 2.902411775371101, 3.499639524681135, 4.053665426600552, 4.454027452905297, 5.151006590442446, 5.967653352318863, 6.152908063276028, 6.459440907629924, 7.039190904476509, 7.599976216244607, 8.074891999306370, 8.634039074716019, 9.051369984613982, 9.370764214645245, 10.05478315553848, 10.32298375026147, 10.81481033940600, 11.31998030541539, 11.75350553436348, 12.03528230772609, 12.50487983397450

Graph of the ZZ-function along the critical line