Properties

Label 2-154-1.1-c3-0-4
Degree $2$
Conductor $154$
Sign $1$
Analytic cond. $9.08629$
Root an. cond. $3.01434$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 2·3-s + 4·4-s + 18·5-s − 4·6-s + 7·7-s + 8·8-s − 23·9-s + 36·10-s − 11·11-s − 8·12-s + 56·13-s + 14·14-s − 36·15-s + 16·16-s + 36·17-s − 46·18-s − 28·19-s + 72·20-s − 14·21-s − 22·22-s + 180·23-s − 16·24-s + 199·25-s + 112·26-s + 100·27-s + 28·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.384·3-s + 1/2·4-s + 1.60·5-s − 0.272·6-s + 0.377·7-s + 0.353·8-s − 0.851·9-s + 1.13·10-s − 0.301·11-s − 0.192·12-s + 1.19·13-s + 0.267·14-s − 0.619·15-s + 1/4·16-s + 0.513·17-s − 0.602·18-s − 0.338·19-s + 0.804·20-s − 0.145·21-s − 0.213·22-s + 1.63·23-s − 0.136·24-s + 1.59·25-s + 0.844·26-s + 0.712·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(154\)    =    \(2 \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(9.08629\)
Root analytic conductor: \(3.01434\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{154} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 154,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.917577395\)
\(L(\frac12)\) \(\approx\) \(2.917577395\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p T \)
7 \( 1 - p T \)
11 \( 1 + p T \)
good3 \( 1 + 2 T + p^{3} T^{2} \)
5 \( 1 - 18 T + p^{3} T^{2} \)
13 \( 1 - 56 T + p^{3} T^{2} \)
17 \( 1 - 36 T + p^{3} T^{2} \)
19 \( 1 + 28 T + p^{3} T^{2} \)
23 \( 1 - 180 T + p^{3} T^{2} \)
29 \( 1 + 54 T + p^{3} T^{2} \)
31 \( 1 + 334 T + p^{3} T^{2} \)
37 \( 1 - 386 T + p^{3} T^{2} \)
41 \( 1 + 444 T + p^{3} T^{2} \)
43 \( 1 + 316 T + p^{3} T^{2} \)
47 \( 1 + 402 T + p^{3} T^{2} \)
53 \( 1 + 486 T + p^{3} T^{2} \)
59 \( 1 + 282 T + p^{3} T^{2} \)
61 \( 1 - 380 T + p^{3} T^{2} \)
67 \( 1 - 176 T + p^{3} T^{2} \)
71 \( 1 + 324 T + p^{3} T^{2} \)
73 \( 1 - 800 T + p^{3} T^{2} \)
79 \( 1 + 1144 T + p^{3} T^{2} \)
83 \( 1 - 468 T + p^{3} T^{2} \)
89 \( 1 + 870 T + p^{3} T^{2} \)
97 \( 1 + 1330 T + p^{3} T^{2} \)
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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

−12.85426914036039494247438562000, −11.33913859448930593525837927929, −10.79568861849419599514645525730, −9.534750594954137451114633023171, −8.398701841336842039676276828045, −6.69353219345719681575517160956, −5.76103708587324493429656248640, −5.09979767709534230915331312883, −3.12808284505645228516634854995, −1.61040963792843664155491799353, 1.61040963792843664155491799353, 3.12808284505645228516634854995, 5.09979767709534230915331312883, 5.76103708587324493429656248640, 6.69353219345719681575517160956, 8.398701841336842039676276828045, 9.534750594954137451114633023171, 10.79568861849419599514645525730, 11.33913859448930593525837927929, 12.85426914036039494247438562000

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