↑

Properties

Label 2-4598-1.1-c1-0-102
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 2.79·3-s + 4-s + 4.02·5-s − 2.79·6-s − 0.274·7-s − 8-s + 4.79·9-s − 4.02·10-s + 2.79·12-s − 0.597·13-s + 0.274·14-s + 11.2·15-s + 16-s + 2.62·17-s − 4.79·18-s + 19-s + 4.02·20-s − 0.767·21-s + 4.04·23-s − 2.79·24-s + 11.1·25-s + 0.597·26-s + 4.99·27-s − 0.274·28-s + 7.29·29-s − 11.2·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.61·3-s + 0.5·4-s + 1.79·5-s − 1.13·6-s − 0.103·7-s − 0.353·8-s + 1.59·9-s − 1.27·10-s + 0.805·12-s − 0.165·13-s + 0.0734·14-s + 2.89·15-s + 0.250·16-s + 0.637·17-s − 1.12·18-s + 0.229·19-s + 0.899·20-s − 0.167·21-s + 0.843·23-s − 0.569·24-s + 2.23·25-s + 0.117·26-s + 0.962·27-s − 0.0519·28-s + 1.35·29-s − 2.05·30-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 3.8870883013.887088301
L(12)L(\frac12) ≈\approx 3.8870883013.887088301
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 1−2.79T+3T2 1 - 2.79T + 3T^{2}
5 1−4.02T+5T2 1 - 4.02T + 5T^{2}
7 1+0.274T+7T2 1 + 0.274T + 7T^{2}
13 1+0.597T+13T2 1 + 0.597T + 13T^{2}
17 1−2.62T+17T2 1 - 2.62T + 17T^{2}
23 1−4.04T+23T2 1 - 4.04T + 23T^{2}
29 1−7.29T+29T2 1 - 7.29T + 29T^{2}
31 1+1.48T+31T2 1 + 1.48T + 31T^{2}
37 1+8.31T+37T2 1 + 8.31T + 37T^{2}
41 1−12.3T+41T2 1 - 12.3T + 41T^{2}
43 1+5.58T+43T2 1 + 5.58T + 43T^{2}
47 1+9.10T+47T2 1 + 9.10T + 47T^{2}
53 1−7.30T+53T2 1 - 7.30T + 53T^{2}
59 1+7.53T+59T2 1 + 7.53T + 59T^{2}
61 1+4.83T+61T2 1 + 4.83T + 61T^{2}
67 1+14.0T+67T2 1 + 14.0T + 67T^{2}
71 1+9.61T+71T2 1 + 9.61T + 71T^{2}
73 1−3.15T+73T2 1 - 3.15T + 73T^{2}
79 1−2.42T+79T2 1 - 2.42T + 79T^{2}
83 1+15.7T+83T2 1 + 15.7T + 83T^{2}
89 1−1.49T+89T2 1 - 1.49T + 89T^{2}
97 1−9.00T+97T2 1 - 9.00T + 97T^{2}
show more
show less
   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

−8.542059510862506552252505549477, −7.76643546484801317608759687461, −7.03375446565863422671825031967, −6.33602629270980713671825421091, −5.50160891405612213977670471675, −4.59355905372068623433634471182, −3.16725376968171598884815843166, −2.85062265538887166126993282975, −1.89875889866990554619032283438, −1.28502238627651904527915799619, 1.28502238627651904527915799619, 1.89875889866990554619032283438, 2.85062265538887166126993282975, 3.16725376968171598884815843166, 4.59355905372068623433634471182, 5.50160891405612213977670471675, 6.33602629270980713671825421091, 7.03375446565863422671825031967, 7.76643546484801317608759687461, 8.542059510862506552252505549477

Graph of the ZZ-function along the critical line