Properties

Label 2-270802-1.1-c1-0-9
Degree $2$
Conductor $270802$
Sign $1$
Analytic cond. $2162.36$
Root an. cond. $46.5012$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2·3-s + 4-s − 2·5-s + 2·6-s + 7-s + 8-s + 9-s − 2·10-s + 2·12-s + 2·13-s + 14-s − 4·15-s + 16-s + 18-s + 6·19-s − 2·20-s + 2·21-s + 23-s + 2·24-s − 25-s + 2·26-s − 4·27-s + 28-s − 4·30-s + 2·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.894·5-s + 0.816·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.577·12-s + 0.554·13-s + 0.267·14-s − 1.03·15-s + 1/4·16-s + 0.235·18-s + 1.37·19-s − 0.447·20-s + 0.436·21-s + 0.208·23-s + 0.408·24-s − 1/5·25-s + 0.392·26-s − 0.769·27-s + 0.188·28-s − 0.730·30-s + 0.359·31-s + 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 270802 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270802 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(270802\)    =    \(2 \cdot 7 \cdot 23 \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(2162.36\)
Root analytic conductor: \(46.5012\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{270802} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 270802,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.915005672\)
\(L(\frac12)\) \(\approx\) \(5.915005672\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
7 \( 1 - T \)
23 \( 1 - T \)
29 \( 1 \)
good3 \( 1 - 2 T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−13.02715524639490, −12.13010916532801, −11.92936173182415, −11.59615414333065, −11.07547503120266, −10.55400258076135, −9.968770639200094, −9.562001407564212, −8.793558783675909, −8.607826461825805, −8.082065793189827, −7.695449439902018, −7.044836894170600, −7.005612331145003, −6.017300391516022, −5.491623050702074, −5.184246891609659, −4.305618045794907, −4.045101282341845, −3.518989440118762, −3.037708542418426, −2.673366350146771, −1.854861406815499, −1.375635606146668, −0.5226586218241309, 0.5226586218241309, 1.375635606146668, 1.854861406815499, 2.673366350146771, 3.037708542418426, 3.518989440118762, 4.045101282341845, 4.305618045794907, 5.184246891609659, 5.491623050702074, 6.017300391516022, 7.005612331145003, 7.044836894170600, 7.695449439902018, 8.082065793189827, 8.607826461825805, 8.793558783675909, 9.562001407564212, 9.968770639200094, 10.55400258076135, 11.07547503120266, 11.59615414333065, 11.92936173182415, 12.13010916532801, 13.02715524639490

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