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Properties

Label 2-1875-1.1-c3-0-210
Degree 22
Conductor 18751875
Sign −1-1
Analytic cond. 110.628110.628
Root an. cond. 10.518010.5180
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.93·2-s + 3·3-s + 0.630·4-s − 8.81·6-s + 18.9·7-s + 21.6·8-s + 9·9-s − 6.06·11-s + 1.89·12-s + 78.2·13-s − 55.5·14-s − 68.6·16-s + 38.4·17-s − 26.4·18-s − 92.9·19-s + 56.7·21-s + 17.8·22-s − 81.0·23-s + 64.9·24-s − 229.·26-s + 27·27-s + 11.9·28-s − 11.1·29-s − 223.·31-s + 28.4·32-s − 18.2·33-s − 112.·34-s + ⋯
L(s)  = 1  − 1.03·2-s + 0.577·3-s + 0.0788·4-s − 0.599·6-s + 1.02·7-s + 0.956·8-s + 0.333·9-s − 0.166·11-s + 0.0455·12-s + 1.66·13-s − 1.06·14-s − 1.07·16-s + 0.547·17-s − 0.346·18-s − 1.12·19-s + 0.589·21-s + 0.172·22-s − 0.734·23-s + 0.552·24-s − 1.73·26-s + 0.192·27-s + 0.0805·28-s − 0.0714·29-s − 1.29·31-s + 0.157·32-s − 0.0960·33-s − 0.569·34-s + ⋯

Functional equation

Λ(s)=(1875s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1875s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18751875    =    3⋅543 \cdot 5^{4}
Sign: −1-1
Analytic conductor: 110.628110.628
Root analytic conductor: 10.518010.5180
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1875, ( :3/2), −1)(2,\ 1875,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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)
bad3 1−3T 1 - 3T
5 1 1
good2 1+2.93T+8T2 1 + 2.93T + 8T^{2}
7 1−18.9T+343T2 1 - 18.9T + 343T^{2}
11 1+6.06T+1.33e3T2 1 + 6.06T + 1.33e3T^{2}
13 1−78.2T+2.19e3T2 1 - 78.2T + 2.19e3T^{2}
17 1−38.4T+4.91e3T2 1 - 38.4T + 4.91e3T^{2}
19 1+92.9T+6.85e3T2 1 + 92.9T + 6.85e3T^{2}
23 1+81.0T+1.21e4T2 1 + 81.0T + 1.21e4T^{2}
29 1+11.1T+2.43e4T2 1 + 11.1T + 2.43e4T^{2}
31 1+223.T+2.97e4T2 1 + 223.T + 2.97e4T^{2}
37 1+107.T+5.06e4T2 1 + 107.T + 5.06e4T^{2}
41 1+350.T+6.89e4T2 1 + 350.T + 6.89e4T^{2}
43 1+356.T+7.95e4T2 1 + 356.T + 7.95e4T^{2}
47 1+291.T+1.03e5T2 1 + 291.T + 1.03e5T^{2}
53 1−684.T+1.48e5T2 1 - 684.T + 1.48e5T^{2}
59 1+223.T+2.05e5T2 1 + 223.T + 2.05e5T^{2}
61 1+697.T+2.26e5T2 1 + 697.T + 2.26e5T^{2}
67 1+909.T+3.00e5T2 1 + 909.T + 3.00e5T^{2}
71 1−201.T+3.57e5T2 1 - 201.T + 3.57e5T^{2}
73 1+291.T+3.89e5T2 1 + 291.T + 3.89e5T^{2}
79 1−709.T+4.93e5T2 1 - 709.T + 4.93e5T^{2}
83 1+1.10e3T+5.71e5T2 1 + 1.10e3T + 5.71e5T^{2}
89 1−3.77T+7.04e5T2 1 - 3.77T + 7.04e5T^{2}
97 1−18.4T+9.12e5T2 1 - 18.4T + 9.12e5T^{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

−8.552524288402296319920664915730, −8.036344299624704961188254129486, −7.28512774116924520580698225083, −6.24817942923156069020841561334, −5.17173263570908966620922184695, −4.24233438986693811896093897335, −3.43790659197265624820137947886, −1.86023140571303293097269545125, −1.42202897673161948304178206609, 0, 1.42202897673161948304178206609, 1.86023140571303293097269545125, 3.43790659197265624820137947886, 4.24233438986693811896093897335, 5.17173263570908966620922184695, 6.24817942923156069020841561334, 7.28512774116924520580698225083, 8.036344299624704961188254129486, 8.552524288402296319920664915730

Graph of the ZZ-function along the critical line