Properties

Label 2-156-1.1-c1-0-0
Degree $2$
Conductor $156$
Sign $1$
Analytic cond. $1.24566$
Root an. cond. $1.11609$
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  + 3-s + 2·7-s + 9-s + 13-s − 6·17-s + 2·19-s + 2·21-s − 5·25-s + 27-s − 6·29-s + 2·31-s + 2·37-s + 39-s − 12·41-s − 4·43-s − 3·49-s − 6·51-s + 6·53-s + 2·57-s + 12·59-s + 2·61-s + 2·63-s − 10·67-s + 12·71-s + 14·73-s − 5·75-s + 8·79-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.755·7-s + 1/3·9-s + 0.277·13-s − 1.45·17-s + 0.458·19-s + 0.436·21-s − 25-s + 0.192·27-s − 1.11·29-s + 0.359·31-s + 0.328·37-s + 0.160·39-s − 1.87·41-s − 0.609·43-s − 3/7·49-s − 0.840·51-s + 0.824·53-s + 0.264·57-s + 1.56·59-s + 0.256·61-s + 0.251·63-s − 1.22·67-s + 1.42·71-s + 1.63·73-s − 0.577·75-s + 0.900·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $1$
Analytic conductor: \(1.24566\)
Root analytic conductor: \(1.11609\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{156} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.376149593\)
\(L(\frac12)\) \(\approx\) \(1.376149593\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
13 \( 1 - T \)
good5 \( 1 + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−13.21941002116462755470673769512, −11.83258307695703192104822918116, −11.02368824085056898890395801496, −9.798663464890123710839719924879, −8.735814680045936069504206621070, −7.86871640882441783507560416572, −6.65843891177652373785895186115, −5.11454570698520102977754794044, −3.80067854107620357152276821608, −2.03881758362108754018941589683, 2.03881758362108754018941589683, 3.80067854107620357152276821608, 5.11454570698520102977754794044, 6.65843891177652373785895186115, 7.86871640882441783507560416572, 8.735814680045936069504206621070, 9.798663464890123710839719924879, 11.02368824085056898890395801496, 11.83258307695703192104822918116, 13.21941002116462755470673769512

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