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Properties

Label 2-4368-1.1-c1-0-34
Degree 22
Conductor 43684368
Sign 11
Analytic cond. 34.878634.8786
Root an. cond. 5.905815.90581
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  + 3-s + 2.56·5-s − 7-s + 9-s + 2·11-s − 13-s + 2.56·15-s − 3.12·17-s + 6.56·19-s − 21-s + 7.68·23-s + 1.56·25-s + 27-s + 0.561·29-s + 2.56·31-s + 2·33-s − 2.56·35-s − 7.12·37-s − 39-s − 1.12·41-s + 5.43·43-s + 2.56·45-s − 5.68·47-s + 49-s − 3.12·51-s + 4.56·53-s + 5.12·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.14·5-s − 0.377·7-s + 0.333·9-s + 0.603·11-s − 0.277·13-s + 0.661·15-s − 0.757·17-s + 1.50·19-s − 0.218·21-s + 1.60·23-s + 0.312·25-s + 0.192·27-s + 0.104·29-s + 0.460·31-s + 0.348·33-s − 0.432·35-s − 1.17·37-s − 0.160·39-s − 0.175·41-s + 0.829·43-s + 0.381·45-s − 0.829·47-s + 0.142·49-s − 0.437·51-s + 0.626·53-s + 0.690·55-s + ⋯

Functional equation

Λ(s)=(4368s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4368s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 43684368    =    24⋅3⋅7⋅132^{4} \cdot 3 \cdot 7 \cdot 13
Sign: 11
Analytic conductor: 34.878634.8786
Root analytic conductor: 5.905815.90581
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4368, ( :1/2), 1)(2,\ 4368,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.1924323073.192432307
L(12)L(\frac12) ≈\approx 3.1924323073.192432307
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
3 1−T 1 - T
7 1+T 1 + T
13 1+T 1 + T
good5 1−2.56T+5T2 1 - 2.56T + 5T^{2}
11 1−2T+11T2 1 - 2T + 11T^{2}
17 1+3.12T+17T2 1 + 3.12T + 17T^{2}
19 1−6.56T+19T2 1 - 6.56T + 19T^{2}
23 1−7.68T+23T2 1 - 7.68T + 23T^{2}
29 1−0.561T+29T2 1 - 0.561T + 29T^{2}
31 1−2.56T+31T2 1 - 2.56T + 31T^{2}
37 1+7.12T+37T2 1 + 7.12T + 37T^{2}
41 1+1.12T+41T2 1 + 1.12T + 41T^{2}
43 1−5.43T+43T2 1 - 5.43T + 43T^{2}
47 1+5.68T+47T2 1 + 5.68T + 47T^{2}
53 1−4.56T+53T2 1 - 4.56T + 53T^{2}
59 1−3.12T+59T2 1 - 3.12T + 59T^{2}
61 1+6T+61T2 1 + 6T + 61T^{2}
67 1+11.3T+67T2 1 + 11.3T + 67T^{2}
71 1−11.1T+71T2 1 - 11.1T + 71T^{2}
73 1−14.8T+73T2 1 - 14.8T + 73T^{2}
79 1+8.80T+79T2 1 + 8.80T + 79T^{2}
83 1−10.8T+83T2 1 - 10.8T + 83T^{2}
89 1−1.43T+89T2 1 - 1.43T + 89T^{2}
97 1+3.43T+97T2 1 + 3.43T + 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

−8.592031143377707751291020430237, −7.53123188168722198939407970556, −6.89818681012205918023028989733, −6.28203039892872926885278307966, −5.37480572561691179871554656684, −4.74352431333538609399444049399, −3.59691775495615444920228810348, −2.88340205073481354091870599228, −2.01154362202121127135375199172, −1.03569791237203781219273122564, 1.03569791237203781219273122564, 2.01154362202121127135375199172, 2.88340205073481354091870599228, 3.59691775495615444920228810348, 4.74352431333538609399444049399, 5.37480572561691179871554656684, 6.28203039892872926885278307966, 6.89818681012205918023028989733, 7.53123188168722198939407970556, 8.592031143377707751291020430237

Graph of the ZZ-function along the critical line