Properties

Label 2-234-1.1-c1-0-1
Degree $2$
Conductor $234$
Sign $1$
Analytic cond. $1.86849$
Root an. cond. $1.36693$
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 + 4-s − 2·5-s + 4·7-s + 8-s − 2·10-s + 4·11-s + 13-s + 4·14-s + 16-s − 2·17-s − 8·19-s − 2·20-s + 4·22-s − 25-s + 26-s + 4·28-s − 6·29-s − 4·31-s + 32-s − 2·34-s − 8·35-s − 2·37-s − 8·38-s − 2·40-s + 10·41-s + 4·43-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.894·5-s + 1.51·7-s + 0.353·8-s − 0.632·10-s + 1.20·11-s + 0.277·13-s + 1.06·14-s + 1/4·16-s − 0.485·17-s − 1.83·19-s − 0.447·20-s + 0.852·22-s − 1/5·25-s + 0.196·26-s + 0.755·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s − 0.342·34-s − 1.35·35-s − 0.328·37-s − 1.29·38-s − 0.316·40-s + 1.56·41-s + 0.609·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.837507996\)
\(L(\frac12)\) \(\approx\) \(1.837507996\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
13 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−12.06877268824115483606570314322, −11.29067241425549122912863548992, −10.82564350612112779643519160383, −9.025474627597816818050752077333, −8.129127522279784382432147914457, −7.16010307294570363491473287814, −5.93418826319229104786149140241, −4.49666099560716857332474018676, −3.92289433795456230197409104475, −1.87879273921165196191180493374, 1.87879273921165196191180493374, 3.92289433795456230197409104475, 4.49666099560716857332474018676, 5.93418826319229104786149140241, 7.16010307294570363491473287814, 8.129127522279784382432147914457, 9.025474627597816818050752077333, 10.82564350612112779643519160383, 11.29067241425549122912863548992, 12.06877268824115483606570314322

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