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Properties

Label 2-4600-1.1-c1-0-23
Degree 22
Conductor 46004600
Sign 11
Analytic cond. 36.731136.7311
Root an. cond. 6.060626.06062
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  − 2.54·3-s − 0.780·7-s + 3.45·9-s + 5.16·11-s − 3.34·13-s + 6.62·17-s + 6.40·19-s + 1.98·21-s − 23-s − 1.15·27-s − 0.0877·29-s + 3.29·31-s − 13.1·33-s + 4.68·37-s + 8.50·39-s − 5.75·41-s − 2.62·43-s − 5.13·47-s − 6.39·49-s − 16.8·51-s − 8.80·53-s − 16.2·57-s + 2.98·59-s − 2.05·61-s − 2.69·63-s + 3.60·67-s + 2.54·69-s + ⋯
L(s)  = 1  − 1.46·3-s − 0.295·7-s + 1.15·9-s + 1.55·11-s − 0.928·13-s + 1.60·17-s + 1.47·19-s + 0.432·21-s − 0.208·23-s − 0.223·27-s − 0.0162·29-s + 0.591·31-s − 2.28·33-s + 0.769·37-s + 1.36·39-s − 0.899·41-s − 0.400·43-s − 0.748·47-s − 0.912·49-s − 2.35·51-s − 1.20·53-s − 2.15·57-s + 0.388·59-s − 0.263·61-s − 0.339·63-s + 0.440·67-s + 0.305·69-s + ⋯

Functional equation

Λ(s)=(4600s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4600s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 46004600    =    23⋅52⋅232^{3} \cdot 5^{2} \cdot 23
Sign: 11
Analytic conductor: 36.731136.7311
Root analytic conductor: 6.060626.06062
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4600, ( :1/2), 1)(2,\ 4600,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.1984249291.198424929
L(12)L(\frac12) ≈\approx 1.1984249291.198424929
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 1
5 1 1
23 1+T 1 + T
good3 1+2.54T+3T2 1 + 2.54T + 3T^{2}
7 1+0.780T+7T2 1 + 0.780T + 7T^{2}
11 1−5.16T+11T2 1 - 5.16T + 11T^{2}
13 1+3.34T+13T2 1 + 3.34T + 13T^{2}
17 1−6.62T+17T2 1 - 6.62T + 17T^{2}
19 1−6.40T+19T2 1 - 6.40T + 19T^{2}
29 1+0.0877T+29T2 1 + 0.0877T + 29T^{2}
31 1−3.29T+31T2 1 - 3.29T + 31T^{2}
37 1−4.68T+37T2 1 - 4.68T + 37T^{2}
41 1+5.75T+41T2 1 + 5.75T + 41T^{2}
43 1+2.62T+43T2 1 + 2.62T + 43T^{2}
47 1+5.13T+47T2 1 + 5.13T + 47T^{2}
53 1+8.80T+53T2 1 + 8.80T + 53T^{2}
59 1−2.98T+59T2 1 - 2.98T + 59T^{2}
61 1+2.05T+61T2 1 + 2.05T + 61T^{2}
67 1−3.60T+67T2 1 - 3.60T + 67T^{2}
71 1−11.5T+71T2 1 - 11.5T + 71T^{2}
73 1−10.4T+73T2 1 - 10.4T + 73T^{2}
79 1−3.43T+79T2 1 - 3.43T + 79T^{2}
83 1+3.62T+83T2 1 + 3.62T + 83T^{2}
89 1−14.3T+89T2 1 - 14.3T + 89T^{2}
97 1−0.427T+97T2 1 - 0.427T + 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

−8.129457702731393416396594879117, −7.45552843160800443399563019391, −6.58875436385154915911588270858, −6.26952816259784275514031713392, −5.27446112674846903234980429007, −4.94812252217822525324622749802, −3.83312049319569116926907133026, −3.07226877912482856153884325216, −1.52089358559558806857771082624, −0.71483432997993236563125489025, 0.71483432997993236563125489025, 1.52089358559558806857771082624, 3.07226877912482856153884325216, 3.83312049319569116926907133026, 4.94812252217822525324622749802, 5.27446112674846903234980429007, 6.26952816259784275514031713392, 6.58875436385154915911588270858, 7.45552843160800443399563019391, 8.129457702731393416396594879117

Graph of the ZZ-function along the critical line