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Properties

Label 2-370-1.1-c1-0-8
Degree 22
Conductor 370370
Sign 11
Analytic cond. 2.954462.95446
Root an. cond. 1.718851.71885
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 + 2·3-s + 4-s + 5-s + 2·6-s + 1.37·7-s + 8-s + 9-s + 10-s − 3.37·11-s + 2·12-s − 4.74·13-s + 1.37·14-s + 2·15-s + 16-s − 5.37·17-s + 18-s − 2·19-s + 20-s + 2.74·21-s − 3.37·22-s + 6.74·23-s + 2·24-s + 25-s − 4.74·26-s − 4·27-s + 1.37·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 0.5·4-s + 0.447·5-s + 0.816·6-s + 0.518·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s − 1.01·11-s + 0.577·12-s − 1.31·13-s + 0.366·14-s + 0.516·15-s + 0.250·16-s − 1.30·17-s + 0.235·18-s − 0.458·19-s + 0.223·20-s + 0.598·21-s − 0.718·22-s + 1.40·23-s + 0.408·24-s + 0.200·25-s − 0.930·26-s − 0.769·27-s + 0.259·28-s + ⋯

Functional equation

Λ(s)=(370s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(370s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 370370    =    2⋅5⋅372 \cdot 5 \cdot 37
Sign: 11
Analytic conductor: 2.954462.95446
Root analytic conductor: 1.718851.71885
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 370, ( :1/2), 1)(2,\ 370,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.8303377692.830337769
L(12)L(\frac12) ≈\approx 2.8303377692.830337769
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
5 1−T 1 - T
37 1−T 1 - T
good3 1−2T+3T2 1 - 2T + 3T^{2}
7 1−1.37T+7T2 1 - 1.37T + 7T^{2}
11 1+3.37T+11T2 1 + 3.37T + 11T^{2}
13 1+4.74T+13T2 1 + 4.74T + 13T^{2}
17 1+5.37T+17T2 1 + 5.37T + 17T^{2}
19 1+2T+19T2 1 + 2T + 19T^{2}
23 1−6.74T+23T2 1 - 6.74T + 23T^{2}
29 1−8.11T+29T2 1 - 8.11T + 29T^{2}
31 1+2.62T+31T2 1 + 2.62T + 31T^{2}
41 1−5.37T+41T2 1 - 5.37T + 41T^{2}
43 1−7.37T+43T2 1 - 7.37T + 43T^{2}
47 1+8.74T+47T2 1 + 8.74T + 47T^{2}
53 1−1.37T+53T2 1 - 1.37T + 53T^{2}
59 1−12.7T+59T2 1 - 12.7T + 59T^{2}
61 1+5.37T+61T2 1 + 5.37T + 61T^{2}
67 1−4.74T+67T2 1 - 4.74T + 67T^{2}
71 1+6.74T+71T2 1 + 6.74T + 71T^{2}
73 1−8.74T+73T2 1 - 8.74T + 73T^{2}
79 1+4.74T+79T2 1 + 4.74T + 79T^{2}
83 1+0.744T+83T2 1 + 0.744T + 83T^{2}
89 1−10T+89T2 1 - 10T + 89T^{2}
97 1+0.116T+97T2 1 + 0.116T + 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

−11.38722906739153977350080833803, −10.54595895376619615626499094419, −9.480271056268451581901414239945, −8.554390353285712280271927196282, −7.67557411468823720785337737584, −6.70394729365155325435215824550, −5.26930273654908337611692424533, −4.46965523456966547152859814665, −2.86056370193655894657740559744, −2.24525849682385529795497473869, 2.24525849682385529795497473869, 2.86056370193655894657740559744, 4.46965523456966547152859814665, 5.26930273654908337611692424533, 6.70394729365155325435215824550, 7.67557411468823720785337737584, 8.554390353285712280271927196282, 9.480271056268451581901414239945, 10.54595895376619615626499094419, 11.38722906739153977350080833803

Graph of the ZZ-function along the critical line