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Properties

Label 2-3971-1.1-c1-0-129
Degree 22
Conductor 39713971
Sign 11
Analytic cond. 31.708531.7085
Root an. cond. 5.631035.63103
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  + 2.66·2-s − 1.82·3-s + 5.09·4-s − 1.36·5-s − 4.86·6-s + 0.192·7-s + 8.23·8-s + 0.334·9-s − 3.64·10-s + 11-s − 9.29·12-s + 1.93·13-s + 0.512·14-s + 2.49·15-s + 11.7·16-s + 2.15·17-s + 0.891·18-s − 6.96·20-s − 0.351·21-s + 2.66·22-s + 2.66·23-s − 15.0·24-s − 3.12·25-s + 5.16·26-s + 4.86·27-s + 0.979·28-s − 10.1·29-s + ⋯
L(s)  = 1  + 1.88·2-s − 1.05·3-s + 2.54·4-s − 0.611·5-s − 1.98·6-s + 0.0727·7-s + 2.91·8-s + 0.111·9-s − 1.15·10-s + 0.301·11-s − 2.68·12-s + 0.538·13-s + 0.136·14-s + 0.644·15-s + 2.93·16-s + 0.521·17-s + 0.210·18-s − 1.55·20-s − 0.0766·21-s + 0.567·22-s + 0.556·23-s − 3.06·24-s − 0.625·25-s + 1.01·26-s + 0.936·27-s + 0.185·28-s − 1.89·29-s + ⋯

Functional equation

Λ(s)=(3971s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3971 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3971s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3971 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 39713971    =    11⋅19211 \cdot 19^{2}
Sign: 11
Analytic conductor: 31.708531.7085
Root analytic conductor: 5.631035.63103
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3971, ( :1/2), 1)(2,\ 3971,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.3484818814.348481881
L(12)L(\frac12) ≈\approx 4.3484818814.348481881
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)
bad11 1−T 1 - T
19 1 1
good2 1−2.66T+2T2 1 - 2.66T + 2T^{2}
3 1+1.82T+3T2 1 + 1.82T + 3T^{2}
5 1+1.36T+5T2 1 + 1.36T + 5T^{2}
7 1−0.192T+7T2 1 - 0.192T + 7T^{2}
13 1−1.93T+13T2 1 - 1.93T + 13T^{2}
17 1−2.15T+17T2 1 - 2.15T + 17T^{2}
23 1−2.66T+23T2 1 - 2.66T + 23T^{2}
29 1+10.1T+29T2 1 + 10.1T + 29T^{2}
31 1−6.26T+31T2 1 - 6.26T + 31T^{2}
37 1−10.3T+37T2 1 - 10.3T + 37T^{2}
41 1−3.77T+41T2 1 - 3.77T + 41T^{2}
43 1−9.07T+43T2 1 - 9.07T + 43T^{2}
47 1−3.51T+47T2 1 - 3.51T + 47T^{2}
53 1+5.13T+53T2 1 + 5.13T + 53T^{2}
59 1−10.4T+59T2 1 - 10.4T + 59T^{2}
61 1−0.568T+61T2 1 - 0.568T + 61T^{2}
67 1−8.04T+67T2 1 - 8.04T + 67T^{2}
71 1+16.1T+71T2 1 + 16.1T + 71T^{2}
73 1−11.9T+73T2 1 - 11.9T + 73T^{2}
79 1−8.27T+79T2 1 - 8.27T + 79T^{2}
83 1−2.24T+83T2 1 - 2.24T + 83T^{2}
89 1−0.963T+89T2 1 - 0.963T + 89T^{2}
97 1−6.81T+97T2 1 - 6.81T + 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

−7.993891456192566923062625032628, −7.44936185341041237807252170889, −6.55463034759847492080231487169, −5.98492372971616364472475344210, −5.50927878622559866672741885199, −4.68148364560363641971132670745, −4.03585780520763248969998019851, −3.33951021657688304989089571369, −2.32676380962450730497981884878, −0.980506032160636117467575452157, 0.980506032160636117467575452157, 2.32676380962450730497981884878, 3.33951021657688304989089571369, 4.03585780520763248969998019851, 4.68148364560363641971132670745, 5.50927878622559866672741885199, 5.98492372971616364472475344210, 6.55463034759847492080231487169, 7.44936185341041237807252170889, 7.993891456192566923062625032628

Graph of the ZZ-function along the critical line