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Properties

Label 2-4598-1.1-c1-0-11
Degree 22
Conductor 45984598
Sign 11
Analytic cond. 36.715236.7152
Root an. cond. 6.059306.05930
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 − 2.23·3-s + 4-s + 0.618·5-s + 2.23·6-s − 2.85·7-s − 8-s + 2.00·9-s − 0.618·10-s − 2.23·12-s − 1.61·13-s + 2.85·14-s − 1.38·15-s + 16-s + 6.23·17-s − 2.00·18-s − 19-s + 0.618·20-s + 6.38·21-s − 0.236·23-s + 2.23·24-s − 4.61·25-s + 1.61·26-s + 2.23·27-s − 2.85·28-s + 10.4·29-s + 1.38·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.29·3-s + 0.5·4-s + 0.276·5-s + 0.912·6-s − 1.07·7-s − 0.353·8-s + 0.666·9-s − 0.195·10-s − 0.645·12-s − 0.448·13-s + 0.762·14-s − 0.356·15-s + 0.250·16-s + 1.51·17-s − 0.471·18-s − 0.229·19-s + 0.138·20-s + 1.39·21-s − 0.0492·23-s + 0.456·24-s − 0.923·25-s + 0.317·26-s + 0.430·27-s − 0.539·28-s + 1.94·29-s + 0.252·30-s + ⋯

Functional equation

Λ(s)=(4598s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4598s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 45984598    =    2⋅112⋅192 \cdot 11^{2} \cdot 19
Sign: 11
Analytic conductor: 36.715236.7152
Root analytic conductor: 6.059306.05930
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4598, ( :1/2), 1)(2,\ 4598,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.44361356830.4436135683
L(12)L(\frac12) ≈\approx 0.44361356830.4436135683
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
11 1 1
19 1+T 1 + T
good3 1+2.23T+3T2 1 + 2.23T + 3T^{2}
5 1−0.618T+5T2 1 - 0.618T + 5T^{2}
7 1+2.85T+7T2 1 + 2.85T + 7T^{2}
13 1+1.61T+13T2 1 + 1.61T + 13T^{2}
17 1−6.23T+17T2 1 - 6.23T + 17T^{2}
23 1+0.236T+23T2 1 + 0.236T + 23T^{2}
29 1−10.4T+29T2 1 - 10.4T + 29T^{2}
31 1+10.7T+31T2 1 + 10.7T + 31T^{2}
37 1+7.85T+37T2 1 + 7.85T + 37T^{2}
41 1+4.09T+41T2 1 + 4.09T + 41T^{2}
43 1+3.70T+43T2 1 + 3.70T + 43T^{2}
47 1−11.4T+47T2 1 - 11.4T + 47T^{2}
53 1−5.94T+53T2 1 - 5.94T + 53T^{2}
59 1+13.0T+59T2 1 + 13.0T + 59T^{2}
61 1−4.38T+61T2 1 - 4.38T + 61T^{2}
67 1+6.09T+67T2 1 + 6.09T + 67T^{2}
71 1+3.76T+71T2 1 + 3.76T + 71T^{2}
73 1+1.32T+73T2 1 + 1.32T + 73T^{2}
79 1+3T+79T2 1 + 3T + 79T^{2}
83 1+1.09T+83T2 1 + 1.09T + 83T^{2}
89 1+9.18T+89T2 1 + 9.18T + 89T^{2}
97 1−16.4T+97T2 1 - 16.4T + 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

−8.384128100991849732609244420164, −7.36487406471447669889341876599, −6.91171087307511925945520357482, −6.03653597334224017526757929125, −5.70255533127782000756365915216, −4.87835054082976535854434014287, −3.66165213990602140085641009450, −2.84291072239016416493754864037, −1.59023624936314229582331173761, −0.44397210129866027016585612795, 0.44397210129866027016585612795, 1.59023624936314229582331173761, 2.84291072239016416493754864037, 3.66165213990602140085641009450, 4.87835054082976535854434014287, 5.70255533127782000756365915216, 6.03653597334224017526757929125, 6.91171087307511925945520357482, 7.36487406471447669889341876599, 8.384128100991849732609244420164

Graph of the ZZ-function along the critical line