Properties

Label 2-1-1.1-c123-0-2
Degree $2$
Conductor $1$
Sign $1$
Analytic cond. $95.8076$
Root an. cond. $9.78813$
Motivic weight $123$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.11e18·2-s − 2.22e29·3-s + 1.54e37·4-s − 1.83e43·5-s + 1.13e48·6-s + 3.56e51·7-s − 2.48e55·8-s + 8.34e56·9-s + 9.39e61·10-s − 1.00e63·11-s − 3.44e66·12-s + 2.32e68·13-s − 1.82e70·14-s + 4.08e72·15-s − 3.77e73·16-s + 8.39e75·17-s − 4.26e75·18-s − 5.57e78·19-s − 2.84e80·20-s − 7.91e80·21-s + 5.12e81·22-s + 7.42e83·23-s + 5.52e84·24-s + 2.43e86·25-s − 1.18e87·26-s + 1.05e88·27-s + 5.52e88·28-s + ⋯
L(s)  = 1  − 1.56·2-s − 1.00·3-s + 1.45·4-s − 1.89·5-s + 1.58·6-s + 0.378·7-s − 0.716·8-s + 0.0171·9-s + 2.96·10-s − 0.0902·11-s − 1.46·12-s + 0.722·13-s − 0.593·14-s + 1.91·15-s − 0.333·16-s + 1.78·17-s − 0.0269·18-s − 1.26·19-s − 2.76·20-s − 0.382·21-s + 0.141·22-s + 1.33·23-s + 0.722·24-s + 2.58·25-s − 1.13·26-s + 0.991·27-s + 0.552·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(124-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+61.5) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(95.8076\)
Root analytic conductor: \(9.78813\)
Motivic weight: \(123\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1,\ (\ :123/2),\ 1)\)

Particular Values

\(L(62)\) \(\approx\) \(0.2685144498\)
\(L(\frac12)\) \(\approx\) \(0.2685144498\)
\(L(\frac{125}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 + 5.11e18T + 1.06e37T^{2} \)
3 \( 1 + 2.22e29T + 4.85e58T^{2} \)
5 \( 1 + 1.83e43T + 9.40e85T^{2} \)
7 \( 1 - 3.56e51T + 8.85e103T^{2} \)
11 \( 1 + 1.00e63T + 1.23e128T^{2} \)
13 \( 1 - 2.32e68T + 1.03e137T^{2} \)
17 \( 1 - 8.39e75T + 2.21e151T^{2} \)
19 \( 1 + 5.57e78T + 1.93e157T^{2} \)
23 \( 1 - 7.42e83T + 3.10e167T^{2} \)
29 \( 1 + 7.68e88T + 7.49e179T^{2} \)
31 \( 1 - 2.02e89T + 2.73e183T^{2} \)
37 \( 1 + 1.76e96T + 7.74e192T^{2} \)
41 \( 1 + 1.88e99T + 2.35e198T^{2} \)
43 \( 1 + 3.07e100T + 8.25e200T^{2} \)
47 \( 1 - 5.66e102T + 4.65e205T^{2} \)
53 \( 1 + 1.10e106T + 1.21e212T^{2} \)
59 \( 1 - 3.85e108T + 6.52e217T^{2} \)
61 \( 1 + 4.04e109T + 3.94e219T^{2} \)
67 \( 1 + 1.24e112T + 4.04e224T^{2} \)
71 \( 1 - 1.34e112T + 5.06e227T^{2} \)
73 \( 1 - 4.67e114T + 1.54e229T^{2} \)
79 \( 1 - 4.47e116T + 2.55e233T^{2} \)
83 \( 1 - 4.53e117T + 1.11e236T^{2} \)
89 \( 1 - 7.57e119T + 5.95e239T^{2} \)
97 \( 1 + 7.20e121T + 2.36e244T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.21970048649010052030687877841, −11.27742219502101323515593303107, −10.58572140682773250564054654648, −8.628860815743764911133457979787, −7.87244789692797398811526312082, −6.71007871955247655252625671818, −4.90098862884172755402579011142, −3.35396501599866560819956972278, −1.24237823812398586002638859057, −0.39607521241551971218488887288, 0.39607521241551971218488887288, 1.24237823812398586002638859057, 3.35396501599866560819956972278, 4.90098862884172755402579011142, 6.71007871955247655252625671818, 7.87244789692797398811526312082, 8.628860815743764911133457979787, 10.58572140682773250564054654648, 11.27742219502101323515593303107, 12.21970048649010052030687877841

Graph of the $Z$-function along the critical line