Properties

Label 2-270802-1.1-c1-0-18
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 $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 7-s + 8-s − 3·9-s + 4·11-s − 4·13-s − 14-s + 16-s + 4·17-s − 3·18-s + 4·19-s + 4·22-s + 23-s − 5·25-s − 4·26-s − 28-s + 2·31-s + 32-s + 4·34-s − 3·36-s + 10·37-s + 4·38-s − 4·41-s − 4·43-s + 4·44-s + 46-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 9-s + 1.20·11-s − 1.10·13-s − 0.267·14-s + 1/4·16-s + 0.970·17-s − 0.707·18-s + 0.917·19-s + 0.852·22-s + 0.208·23-s − 25-s − 0.784·26-s − 0.188·28-s + 0.359·31-s + 0.176·32-s + 0.685·34-s − 1/2·36-s + 1.64·37-s + 0.648·38-s − 0.624·41-s − 0.609·43-s + 0.603·44-s + 0.147·46-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: \(1\)
Selberg data: \((2,\ 270802,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 10 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.15424381443803, −12.36393125739370, −12.00477875696299, −11.74737045333800, −11.54586812295480, −10.70273019473275, −10.32092425663931, −9.674034899120313, −9.416199410528348, −8.965637077671546, −8.248561465857564, −7.744120081515280, −7.371638868509632, −6.847000941151948, −6.243519584640795, −5.811887504819426, −5.532583516600031, −4.805498487104513, −4.407165821671253, −3.684857043409039, −3.342170312457697, −2.738729379419118, −2.328503226187867, −1.461586759779826, −0.8858093167989513, 0, 0.8858093167989513, 1.461586759779826, 2.328503226187867, 2.738729379419118, 3.342170312457697, 3.684857043409039, 4.407165821671253, 4.805498487104513, 5.532583516600031, 5.811887504819426, 6.243519584640795, 6.847000941151948, 7.371638868509632, 7.744120081515280, 8.248561465857564, 8.965637077671546, 9.416199410528348, 9.674034899120313, 10.32092425663931, 10.70273019473275, 11.54586812295480, 11.74737045333800, 12.00477875696299, 12.36393125739370, 13.15424381443803

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