Properties

Label 2-10-1.1-c19-0-0
Degree $2$
Conductor $10$
Sign $1$
Analytic cond. $22.8816$
Root an. cond. $4.78347$
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  − 512·2-s − 2.66e4·3-s + 2.62e5·4-s − 1.95e6·5-s + 1.36e7·6-s − 3.98e7·7-s − 1.34e8·8-s − 4.53e8·9-s + 1.00e9·10-s − 1.01e10·11-s − 6.97e9·12-s − 2.69e10·13-s + 2.04e10·14-s + 5.19e10·15-s + 6.87e10·16-s − 8.01e10·17-s + 2.32e11·18-s − 1.16e12·19-s − 5.12e11·20-s + 1.06e12·21-s + 5.20e12·22-s + 1.37e13·23-s + 3.57e12·24-s + 3.81e12·25-s + 1.38e13·26-s + 4.30e13·27-s − 1.04e13·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.780·3-s + 1/2·4-s − 0.447·5-s + 0.552·6-s − 0.373·7-s − 0.353·8-s − 0.390·9-s + 0.316·10-s − 1.29·11-s − 0.390·12-s − 0.705·13-s + 0.264·14-s + 0.349·15-s + 1/4·16-s − 0.163·17-s + 0.275·18-s − 0.831·19-s − 0.223·20-s + 0.291·21-s + 0.918·22-s + 1.59·23-s + 0.276·24-s + 1/5·25-s + 0.498·26-s + 1.08·27-s − 0.186·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $1$
Analytic conductor: \(22.8816\)
Root analytic conductor: \(4.78347\)
Motivic weight: \(19\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :19/2),\ 1)\)

Particular Values

\(L(10)\) \(\approx\) \(0.4186117364\)
\(L(\frac12)\) \(\approx\) \(0.4186117364\)
\(L(\frac{21}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p^{9} T \)
5 \( 1 + p^{9} T \)
good3 \( 1 + 986 p^{3} T + p^{19} T^{2} \)
7 \( 1 + 5697718 p T + p^{19} T^{2} \)
11 \( 1 + 10161579168 T + p^{19} T^{2} \)
13 \( 1 + 26970649702 T + p^{19} T^{2} \)
17 \( 1 + 4714985478 p T + p^{19} T^{2} \)
19 \( 1 + 1169772071260 T + p^{19} T^{2} \)
23 \( 1 - 13795883851698 T + p^{19} T^{2} \)
29 \( 1 - 65324757765390 T + p^{19} T^{2} \)
31 \( 1 + 8926539984748 T + p^{19} T^{2} \)
37 \( 1 - 525454617064394 T + p^{19} T^{2} \)
41 \( 1 + 2635226882131818 T + p^{19} T^{2} \)
43 \( 1 + 1501708702325062 T + p^{19} T^{2} \)
47 \( 1 + 3651608570665986 T + p^{19} T^{2} \)
53 \( 1 - 43306800238889538 T + p^{19} T^{2} \)
59 \( 1 - 51652090463616180 T + p^{19} T^{2} \)
61 \( 1 - 45200043953043002 T + p^{19} T^{2} \)
67 \( 1 - 322077213275888894 T + p^{19} T^{2} \)
71 \( 1 - 393293311705873692 T + p^{19} T^{2} \)
73 \( 1 + 672469661893471342 T + p^{19} T^{2} \)
79 \( 1 + 482639101471927720 T + p^{19} T^{2} \)
83 \( 1 + 313265345629507302 T + p^{19} T^{2} \)
89 \( 1 + 4230101056729722390 T + p^{19} T^{2} \)
97 \( 1 - 354705113301714434 T + p^{19} T^{2} \)
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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

−16.38560862496191299395205572834, −15.08433016440074855592044410122, −12.83140715066330016793430642244, −11.41460281961255170653038122196, −10.25807833409575336672602977002, −8.422818414744066292091225133853, −6.83000332307881701260337071238, −5.14145694070430837143794016731, −2.73373787560914114168136793491, −0.47231849845202274310640815926, 0.47231849845202274310640815926, 2.73373787560914114168136793491, 5.14145694070430837143794016731, 6.83000332307881701260337071238, 8.422818414744066292091225133853, 10.25807833409575336672602977002, 11.41460281961255170653038122196, 12.83140715066330016793430642244, 15.08433016440074855592044410122, 16.38560862496191299395205572834

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