Properties

Label 2-1-1.1-c29-0-0
Degree $2$
Conductor $1$
Sign $-1$
Analytic cond. $5.32780$
Root an. cond. $2.30820$
Motivic weight $29$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.74e4·2-s + 9.52e6·3-s − 2.32e8·4-s − 1.12e10·5-s − 1.66e11·6-s − 2.98e12·7-s + 1.34e13·8-s + 2.20e13·9-s + 1.96e14·10-s − 1.25e15·11-s − 2.21e15·12-s + 6.09e15·13-s + 5.20e16·14-s − 1.07e17·15-s − 1.08e17·16-s + 8.28e16·17-s − 3.84e17·18-s + 3.72e18·19-s + 2.62e18·20-s − 2.84e19·21-s + 2.19e19·22-s − 1.69e19·23-s + 1.27e20·24-s − 5.94e19·25-s − 1.06e20·26-s − 4.43e20·27-s + 6.95e20·28-s + ⋯
L(s)  = 1  − 0.752·2-s + 1.14·3-s − 0.433·4-s − 0.825·5-s − 0.865·6-s − 1.66·7-s + 1.07·8-s + 0.321·9-s + 0.620·10-s − 0.998·11-s − 0.498·12-s + 0.429·13-s + 1.25·14-s − 0.948·15-s − 0.377·16-s + 0.119·17-s − 0.242·18-s + 1.06·19-s + 0.358·20-s − 1.91·21-s + 0.751·22-s − 0.304·23-s + 1.24·24-s − 0.319·25-s − 0.322·26-s − 0.779·27-s + 0.722·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(30-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+29/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $-1$
Analytic conductor: \(5.32780\)
Root analytic conductor: \(2.30820\)
Motivic weight: \(29\)
Rational: no
Arithmetic: yes
Character: $\chi_{1} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1,\ (\ :29/2),\ -1)\)

Particular Values

\(L(15)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{31}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 + 1.74e4T + 5.36e8T^{2} \)
3 \( 1 - 9.52e6T + 6.86e13T^{2} \)
5 \( 1 + 1.12e10T + 1.86e20T^{2} \)
7 \( 1 + 2.98e12T + 3.21e24T^{2} \)
11 \( 1 + 1.25e15T + 1.58e30T^{2} \)
13 \( 1 - 6.09e15T + 2.01e32T^{2} \)
17 \( 1 - 8.28e16T + 4.81e35T^{2} \)
19 \( 1 - 3.72e18T + 1.21e37T^{2} \)
23 \( 1 + 1.69e19T + 3.09e39T^{2} \)
29 \( 1 + 8.81e20T + 2.56e42T^{2} \)
31 \( 1 - 3.71e21T + 1.77e43T^{2} \)
37 \( 1 - 1.97e22T + 3.00e45T^{2} \)
41 \( 1 + 1.00e23T + 5.89e46T^{2} \)
43 \( 1 + 1.61e23T + 2.34e47T^{2} \)
47 \( 1 + 2.61e24T + 3.09e48T^{2} \)
53 \( 1 + 1.44e25T + 1.00e50T^{2} \)
59 \( 1 - 5.87e24T + 2.26e51T^{2} \)
61 \( 1 + 6.18e25T + 5.95e51T^{2} \)
67 \( 1 - 1.29e26T + 9.04e52T^{2} \)
71 \( 1 + 3.63e26T + 4.85e53T^{2} \)
73 \( 1 - 1.45e27T + 1.08e54T^{2} \)
79 \( 1 + 3.93e27T + 1.07e55T^{2} \)
83 \( 1 - 6.61e27T + 4.50e55T^{2} \)
89 \( 1 - 6.47e27T + 3.40e56T^{2} \)
97 \( 1 - 9.52e28T + 4.13e57T^{2} \)
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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

−25.72868657660485147030428912722, −22.91132095957644566528577857099, −19.93377438460368859215936465405, −18.82169509011267587663733755437, −15.92917042077896086436932580822, −13.38692374908591032596541219977, −9.637249782268918632286389013119, −7.949866560858724884326906739463, −3.34698244840578429721006084056, 0, 3.34698244840578429721006084056, 7.949866560858724884326906739463, 9.637249782268918632286389013119, 13.38692374908591032596541219977, 15.92917042077896086436932580822, 18.82169509011267587663733755437, 19.93377438460368859215936465405, 22.91132095957644566528577857099, 25.72868657660485147030428912722

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