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Properties

Label 2-3750-1.1-c1-0-12
Degree 22
Conductor 37503750
Sign 11
Analytic cond. 29.943929.9439
Root an. cond. 5.472105.47210
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-s + 3-s + 4-s − 6-s − 0.547·7-s − 8-s + 9-s − 2.03·11-s + 12-s + 1.67·13-s + 0.547·14-s + 16-s − 0.445·17-s − 18-s + 0.0854·19-s − 0.547·21-s + 2.03·22-s + 7.70·23-s − 24-s − 1.67·26-s + 27-s − 0.547·28-s + 1.55·29-s − 7.53·31-s − 32-s − 2.03·33-s + 0.445·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s − 0.206·7-s − 0.353·8-s + 0.333·9-s − 0.613·11-s + 0.288·12-s + 0.464·13-s + 0.146·14-s + 0.250·16-s − 0.107·17-s − 0.235·18-s + 0.0196·19-s − 0.119·21-s + 0.434·22-s + 1.60·23-s − 0.204·24-s − 0.328·26-s + 0.192·27-s − 0.103·28-s + 0.289·29-s − 1.35·31-s − 0.176·32-s − 0.354·33-s + 0.0763·34-s + ⋯

Functional equation

Λ(s)=(3750s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3750s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 37503750    =    2⋅3⋅542 \cdot 3 \cdot 5^{4}
Sign: 11
Analytic conductor: 29.943929.9439
Root analytic conductor: 5.472105.47210
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3750, ( :1/2), 1)(2,\ 3750,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.5973969091.597396909
L(12)L(\frac12) ≈\approx 1.5973969091.597396909
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
good7 1+0.547T+7T2 1 + 0.547T + 7T^{2}
11 1+2.03T+11T2 1 + 2.03T + 11T^{2}
13 1−1.67T+13T2 1 - 1.67T + 13T^{2}
17 1+0.445T+17T2 1 + 0.445T + 17T^{2}
19 1−0.0854T+19T2 1 - 0.0854T + 19T^{2}
23 1−7.70T+23T2 1 - 7.70T + 23T^{2}
29 1−1.55T+29T2 1 - 1.55T + 29T^{2}
31 1+7.53T+31T2 1 + 7.53T + 31T^{2}
37 1−0.0660T+37T2 1 - 0.0660T + 37T^{2}
41 1−8.80T+41T2 1 - 8.80T + 41T^{2}
43 1+2.00T+43T2 1 + 2.00T + 43T^{2}
47 1−7.22T+47T2 1 - 7.22T + 47T^{2}
53 1−1.98T+53T2 1 - 1.98T + 53T^{2}
59 1−5.87T+59T2 1 - 5.87T + 59T^{2}
61 1+12.1T+61T2 1 + 12.1T + 61T^{2}
67 1−6.58T+67T2 1 - 6.58T + 67T^{2}
71 1+8.05T+71T2 1 + 8.05T + 71T^{2}
73 1−10.5T+73T2 1 - 10.5T + 73T^{2}
79 1−7.37T+79T2 1 - 7.37T + 79T^{2}
83 1+1.45T+83T2 1 + 1.45T + 83T^{2}
89 1−9.44T+89T2 1 - 9.44T + 89T^{2}
97 1−6.58T+97T2 1 - 6.58T + 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

−8.654402911564188064849023467221, −7.78268170716965157721233235542, −7.28255068864731722673691217842, −6.49087490561983024831756652715, −5.61332673687601167498700471768, −4.72213401282573685579241742510, −3.61553395258646265915848371339, −2.88556905671616012570440214887, −1.97866004340215080045084800956, −0.796807578777940509871175178271, 0.796807578777940509871175178271, 1.97866004340215080045084800956, 2.88556905671616012570440214887, 3.61553395258646265915848371339, 4.72213401282573685579241742510, 5.61332673687601167498700471768, 6.49087490561983024831756652715, 7.28255068864731722673691217842, 7.78268170716965157721233235542, 8.654402911564188064849023467221

Graph of the ZZ-function along the critical line