Properties

Label 2-1-1.1-c53-0-1
Degree $2$
Conductor $1$
Sign $-1$
Analytic cond. $17.7903$
Root an. cond. $4.21785$
Motivic weight $53$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.58e7·2-s − 2.97e12·3-s − 8.33e15·4-s + 5.38e18·5-s − 7.69e19·6-s + 2.33e22·7-s − 4.48e23·8-s − 1.05e25·9-s + 1.39e26·10-s − 3.50e27·11-s + 2.48e28·12-s − 3.76e28·13-s + 6.03e29·14-s − 1.60e31·15-s + 6.35e31·16-s − 5.80e32·17-s − 2.71e32·18-s − 9.40e33·19-s − 4.48e34·20-s − 6.95e34·21-s − 9.06e34·22-s − 5.69e35·23-s + 1.33e36·24-s + 1.78e37·25-s − 9.73e35·26-s + 8.90e37·27-s − 1.94e38·28-s + ⋯
L(s)  = 1  + 0.272·2-s − 0.676·3-s − 0.925·4-s + 1.61·5-s − 0.184·6-s + 0.939·7-s − 0.524·8-s − 0.542·9-s + 0.439·10-s − 0.887·11-s + 0.626·12-s − 0.113·13-s + 0.255·14-s − 1.09·15-s + 0.783·16-s − 1.43·17-s − 0.147·18-s − 1.21·19-s − 1.49·20-s − 0.635·21-s − 0.241·22-s − 0.467·23-s + 0.354·24-s + 1.60·25-s − 0.0310·26-s + 1.04·27-s − 0.870·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(54-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+53/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $-1$
Analytic conductor: \(17.7903\)
Root analytic conductor: \(4.21785\)
Motivic weight: \(53\)
Rational: no
Arithmetic: yes
Character: $\chi_{1} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1,\ (\ :53/2),\ -1)\)

Particular Values

\(L(27)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{55}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 - 2.58e7T + 9.00e15T^{2} \)
3 \( 1 + 2.97e12T + 1.93e25T^{2} \)
5 \( 1 - 5.38e18T + 1.11e37T^{2} \)
7 \( 1 - 2.33e22T + 6.16e44T^{2} \)
11 \( 1 + 3.50e27T + 1.56e55T^{2} \)
13 \( 1 + 3.76e28T + 1.09e59T^{2} \)
17 \( 1 + 5.80e32T + 1.63e65T^{2} \)
19 \( 1 + 9.40e33T + 5.94e67T^{2} \)
23 \( 1 + 5.69e35T + 1.48e72T^{2} \)
29 \( 1 + 2.46e38T + 3.21e77T^{2} \)
31 \( 1 + 4.25e39T + 1.10e79T^{2} \)
37 \( 1 + 5.85e41T + 1.30e83T^{2} \)
41 \( 1 - 5.37e42T + 3.00e85T^{2} \)
43 \( 1 - 1.54e43T + 3.74e86T^{2} \)
47 \( 1 + 1.14e44T + 4.18e88T^{2} \)
53 \( 1 - 8.51e44T + 2.43e91T^{2} \)
59 \( 1 + 9.48e46T + 7.16e93T^{2} \)
61 \( 1 - 3.37e47T + 4.19e94T^{2} \)
67 \( 1 + 3.70e48T + 6.05e96T^{2} \)
71 \( 1 - 2.16e48T + 1.30e98T^{2} \)
73 \( 1 + 1.03e49T + 5.70e98T^{2} \)
79 \( 1 - 5.02e49T + 3.75e100T^{2} \)
83 \( 1 + 3.18e50T + 5.14e101T^{2} \)
89 \( 1 - 3.04e51T + 2.07e103T^{2} \)
97 \( 1 - 4.29e52T + 1.99e105T^{2} \)
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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

−17.99666868187723941445779264840, −17.36102932607581509596883384561, −14.36551689731113482282739137020, −13.03731209265050034915417118205, −10.68358842611485570456741833126, −8.833179282512613733816156938047, −5.89849610083662869023730756189, −4.86517752815356537152155760419, −2.08200369414158559766490705768, 0, 2.08200369414158559766490705768, 4.86517752815356537152155760419, 5.89849610083662869023730756189, 8.833179282512613733816156938047, 10.68358842611485570456741833126, 13.03731209265050034915417118205, 14.36551689731113482282739137020, 17.36102932607581509596883384561, 17.99666868187723941445779264840

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