Properties

Label 2-1-1.1-c19-0-0
Degree $2$
Conductor $1$
Sign $1$
Analytic cond. $2.28816$
Root an. cond. $1.51266$
Motivic weight $19$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 456·2-s + 5.06e4·3-s − 3.16e5·4-s − 2.37e6·5-s + 2.30e7·6-s − 1.69e7·7-s − 3.83e8·8-s + 1.40e9·9-s − 1.08e9·10-s − 1.62e7·11-s − 1.60e10·12-s + 5.04e10·13-s − 7.71e9·14-s − 1.20e11·15-s − 8.93e9·16-s + 2.25e11·17-s + 6.39e11·18-s − 1.71e12·19-s + 7.52e11·20-s − 8.56e11·21-s − 7.39e9·22-s + 1.40e13·23-s − 1.94e13·24-s − 1.34e13·25-s + 2.29e13·26-s + 1.22e13·27-s + 5.35e12·28-s + ⋯
L(s)  = 1  + 0.629·2-s + 1.48·3-s − 0.603·4-s − 0.544·5-s + 0.935·6-s − 0.158·7-s − 1.00·8-s + 1.20·9-s − 0.342·10-s − 0.00207·11-s − 0.896·12-s + 1.31·13-s − 0.0997·14-s − 0.808·15-s − 0.0325·16-s + 0.460·17-s + 0.760·18-s − 1.21·19-s + 0.328·20-s − 0.235·21-s − 0.00130·22-s + 1.62·23-s − 1.50·24-s − 0.703·25-s + 0.830·26-s + 0.308·27-s + 0.0956·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(20-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+19/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(2.28816\)
Root analytic conductor: \(1.51266\)
Motivic weight: \(19\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1,\ (\ :19/2),\ 1)\)

Particular Values

\(L(10)\) \(\approx\) \(1.981735405\)
\(L(\frac12)\) \(\approx\) \(1.981735405\)
\(L(\frac{21}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 - 57 p^{3} T + p^{19} T^{2} \)
3 \( 1 - 1876 p^{3} T + p^{19} T^{2} \)
5 \( 1 + 475482 p T + p^{19} T^{2} \)
7 \( 1 + 345256 p^{2} T + p^{19} T^{2} \)
11 \( 1 + 1473828 p T + p^{19} T^{2} \)
13 \( 1 - 3878585774 p T + p^{19} T^{2} \)
17 \( 1 - 13239417618 p T + p^{19} T^{2} \)
19 \( 1 + 1710278572660 T + p^{19} T^{2} \)
23 \( 1 - 14036534788872 T + p^{19} T^{2} \)
29 \( 1 - 1137835269510 T + p^{19} T^{2} \)
31 \( 1 + 104626880141728 T + p^{19} T^{2} \)
37 \( 1 + 169392327370594 T + p^{19} T^{2} \)
41 \( 1 + 3309984750560838 T + p^{19} T^{2} \)
43 \( 1 - 1127913532193492 T + p^{19} T^{2} \)
47 \( 1 - 3498693987674256 T + p^{19} T^{2} \)
53 \( 1 - 29956294112980302 T + p^{19} T^{2} \)
59 \( 1 - 58391397642732420 T + p^{19} T^{2} \)
61 \( 1 - 23373685132672742 T + p^{19} T^{2} \)
67 \( 1 + 205102524257382244 T + p^{19} T^{2} \)
71 \( 1 + 177902341950417768 T + p^{19} T^{2} \)
73 \( 1 - 299853775038660122 T + p^{19} T^{2} \)
79 \( 1 + 92227090144007440 T + p^{19} T^{2} \)
83 \( 1 - 1208542823470585932 T + p^{19} T^{2} \)
89 \( 1 - 4371201192290304330 T + p^{19} T^{2} \)
97 \( 1 + 635013222218448094 T + p^{19} T^{2} \)
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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

−30.55672605125458804006448947936, −27.29445208791780927881588765643, −25.59118621334058836769515346551, −23.37831105449752633139466303810, −21.00993031068581648326307127139, −19.01545399887427581559957963587, −15.00403257750766652703892999605, −13.31064190731384332629695179807, −8.677106764116385551617153434409, −3.60719823716604384253001609923, 3.60719823716604384253001609923, 8.677106764116385551617153434409, 13.31064190731384332629695179807, 15.00403257750766652703892999605, 19.01545399887427581559957963587, 21.00993031068581648326307127139, 23.37831105449752633139466303810, 25.59118621334058836769515346551, 27.29445208791780927881588765643, 30.55672605125458804006448947936

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