Properties

Label 2-1-1.1-c45-0-1
Degree $2$
Conductor $1$
Sign $-1$
Analytic cond. $12.8255$
Root an. cond. $3.58128$
Motivic weight $45$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7.36e6·2-s − 5.75e10·3-s + 1.91e13·4-s + 5.97e15·5-s − 4.23e17·6-s − 1.35e19·7-s − 1.18e20·8-s + 3.55e20·9-s + 4.40e22·10-s − 4.62e23·11-s − 1.09e24·12-s + 1.84e24·13-s − 1.00e26·14-s − 3.43e26·15-s − 1.54e27·16-s + 5.39e27·17-s + 2.62e27·18-s + 2.04e28·19-s + 1.14e29·20-s + 7.81e29·21-s − 3.40e30·22-s − 4.06e30·23-s + 6.81e30·24-s + 7.24e30·25-s + 1.36e31·26-s + 1.49e32·27-s − 2.59e32·28-s + ⋯
L(s)  = 1  + 1.24·2-s − 1.05·3-s + 0.543·4-s + 1.12·5-s − 1.31·6-s − 1.31·7-s − 0.567·8-s + 0.120·9-s + 1.39·10-s − 1.71·11-s − 0.574·12-s + 0.159·13-s − 1.63·14-s − 1.18·15-s − 1.24·16-s + 1.11·17-s + 0.149·18-s + 0.345·19-s + 0.608·20-s + 1.38·21-s − 2.12·22-s − 0.933·23-s + 0.600·24-s + 0.255·25-s + 0.198·26-s + 0.931·27-s − 0.712·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(46-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+45/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $-1$
Analytic conductor: \(12.8255\)
Root analytic conductor: \(3.58128\)
Motivic weight: \(45\)
Rational: no
Arithmetic: yes
Character: $\chi_{1} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1,\ (\ :45/2),\ -1)\)

Particular Values

\(L(23)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{47}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 - 7.36e6T + 3.51e13T^{2} \)
3 \( 1 + 5.75e10T + 2.95e21T^{2} \)
5 \( 1 - 5.97e15T + 2.84e31T^{2} \)
7 \( 1 + 1.35e19T + 1.07e38T^{2} \)
11 \( 1 + 4.62e23T + 7.28e46T^{2} \)
13 \( 1 - 1.84e24T + 1.34e50T^{2} \)
17 \( 1 - 5.39e27T + 2.34e55T^{2} \)
19 \( 1 - 2.04e28T + 3.49e57T^{2} \)
23 \( 1 + 4.06e30T + 1.89e61T^{2} \)
29 \( 1 - 6.91e32T + 6.42e65T^{2} \)
31 \( 1 + 4.29e32T + 1.29e67T^{2} \)
37 \( 1 + 1.38e35T + 3.70e70T^{2} \)
41 \( 1 - 9.02e35T + 3.76e72T^{2} \)
43 \( 1 + 2.22e36T + 3.20e73T^{2} \)
47 \( 1 + 1.00e37T + 1.75e75T^{2} \)
53 \( 1 + 2.47e38T + 3.91e77T^{2} \)
59 \( 1 - 1.01e40T + 4.87e79T^{2} \)
61 \( 1 + 2.35e40T + 2.18e80T^{2} \)
67 \( 1 + 2.39e41T + 1.49e82T^{2} \)
71 \( 1 - 4.61e40T + 2.02e83T^{2} \)
73 \( 1 - 6.89e41T + 7.07e83T^{2} \)
79 \( 1 + 1.80e42T + 2.47e85T^{2} \)
83 \( 1 - 1.09e42T + 2.28e86T^{2} \)
89 \( 1 + 5.72e43T + 5.27e87T^{2} \)
97 \( 1 + 2.85e44T + 2.53e89T^{2} \)
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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

−21.21486090470818123577283595192, −18.06574641877894839779638823471, −16.10130476472056037967319118127, −13.66256273110436947434727107367, −12.43077072814230613599510364838, −10.10487225466447691656417923583, −6.13161181216014691666479320551, −5.33002238531523506864245254531, −2.91839662490823476782717262862, 0, 2.91839662490823476782717262862, 5.33002238531523506864245254531, 6.13161181216014691666479320551, 10.10487225466447691656417923583, 12.43077072814230613599510364838, 13.66256273110436947434727107367, 16.10130476472056037967319118127, 18.06574641877894839779638823471, 21.21486090470818123577283595192

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