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Properties

Label 2-363-1.1-c1-0-13
Degree 22
Conductor 363363
Sign −1-1
Analytic cond. 2.898562.89856
Root an. cond. 1.702511.70251
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.381·2-s − 3-s − 1.85·4-s + 1.61·5-s + 0.381·6-s − 7-s + 1.47·8-s + 9-s − 0.618·10-s + 1.85·12-s − 4.23·13-s + 0.381·14-s − 1.61·15-s + 3.14·16-s − 7.85·17-s − 0.381·18-s − 0.854·19-s − 3·20-s + 21-s − 4.23·23-s − 1.47·24-s − 2.38·25-s + 1.61·26-s − 27-s + 1.85·28-s − 6·29-s + 0.618·30-s + ⋯
L(s)  = 1  − 0.270·2-s − 0.577·3-s − 0.927·4-s + 0.723·5-s + 0.155·6-s − 0.377·7-s + 0.520·8-s + 0.333·9-s − 0.195·10-s + 0.535·12-s − 1.17·13-s + 0.102·14-s − 0.417·15-s + 0.786·16-s − 1.90·17-s − 0.0900·18-s − 0.195·19-s − 0.670·20-s + 0.218·21-s − 0.883·23-s − 0.300·24-s − 0.476·25-s + 0.317·26-s − 0.192·27-s + 0.350·28-s − 1.11·29-s + 0.112·30-s + ⋯

Functional equation

Λ(s)=(363s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(363s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 363363    =    3⋅1123 \cdot 11^{2}
Sign: −1-1
Analytic conductor: 2.898562.89856
Root analytic conductor: 1.702511.70251
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 363, ( :1/2), −1)(2,\ 363,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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)
bad3 1+T 1 + T
11 1 1
good2 1+0.381T+2T2 1 + 0.381T + 2T^{2}
5 1−1.61T+5T2 1 - 1.61T + 5T^{2}
7 1+T+7T2 1 + T + 7T^{2}
13 1+4.23T+13T2 1 + 4.23T + 13T^{2}
17 1+7.85T+17T2 1 + 7.85T + 17T^{2}
19 1+0.854T+19T2 1 + 0.854T + 19T^{2}
23 1+4.23T+23T2 1 + 4.23T + 23T^{2}
29 1+6T+29T2 1 + 6T + 29T^{2}
31 1−5.09T+31T2 1 - 5.09T + 31T^{2}
37 1+1.76T+37T2 1 + 1.76T + 37T^{2}
41 1+4.23T+41T2 1 + 4.23T + 41T^{2}
43 1−6.70T+43T2 1 - 6.70T + 43T^{2}
47 1−1.09T+47T2 1 - 1.09T + 47T^{2}
53 1+2.61T+53T2 1 + 2.61T + 53T^{2}
59 1−9.61T+59T2 1 - 9.61T + 59T^{2}
61 1−8.56T+61T2 1 - 8.56T + 61T^{2}
67 1+4.85T+67T2 1 + 4.85T + 67T^{2}
71 1+5.32T+71T2 1 + 5.32T + 71T^{2}
73 1−7.70T+73T2 1 - 7.70T + 73T^{2}
79 1−11T+79T2 1 - 11T + 79T^{2}
83 1+7.47T+83T2 1 + 7.47T + 83T^{2}
89 1+3.76T+89T2 1 + 3.76T + 89T^{2}
97 1−1.14T+97T2 1 - 1.14T + 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

−10.78289067142785881077653490406, −9.863709486678904235572054203793, −9.396881642356561666798327339601, −8.313693641954583186886394218883, −7.08970735101694332655335795573, −6.05540157654284710868464828254, −5.03013832220991572063030127937, −4.08942123162730063370882791638, −2.14593612553279308138501529502, 0, 2.14593612553279308138501529502, 4.08942123162730063370882791638, 5.03013832220991572063030127937, 6.05540157654284710868464828254, 7.08970735101694332655335795573, 8.313693641954583186886394218883, 9.396881642356561666798327339601, 9.863709486678904235572054203793, 10.78289067142785881077653490406

Graph of the ZZ-function along the critical line