Properties

Label 2-114-1.1-c3-0-6
Degree $2$
Conductor $114$
Sign $-1$
Analytic cond. $6.72621$
Root an. cond. $2.59349$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·3-s + 4·4-s − 7·5-s − 6·6-s − 15·7-s − 8·8-s + 9·9-s + 14·10-s − 49·11-s + 12·12-s + 14·13-s + 30·14-s − 21·15-s + 16·16-s − 33·17-s − 18·18-s − 19·19-s − 28·20-s − 45·21-s + 98·22-s − 148·23-s − 24·24-s − 76·25-s − 28·26-s + 27·27-s − 60·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.626·5-s − 0.408·6-s − 0.809·7-s − 0.353·8-s + 1/3·9-s + 0.442·10-s − 1.34·11-s + 0.288·12-s + 0.298·13-s + 0.572·14-s − 0.361·15-s + 1/4·16-s − 0.470·17-s − 0.235·18-s − 0.229·19-s − 0.313·20-s − 0.467·21-s + 0.949·22-s − 1.34·23-s − 0.204·24-s − 0.607·25-s − 0.211·26-s + 0.192·27-s − 0.404·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(114\)    =    \(2 \cdot 3 \cdot 19\)
Sign: $-1$
Analytic conductor: \(6.72621\)
Root analytic conductor: \(2.59349\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 114,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p T \)
3 \( 1 - p T \)
19 \( 1 + p T \)
good5 \( 1 + 7 T + p^{3} T^{2} \)
7 \( 1 + 15 T + p^{3} T^{2} \)
11 \( 1 + 49 T + p^{3} T^{2} \)
13 \( 1 - 14 T + p^{3} T^{2} \)
17 \( 1 + 33 T + p^{3} T^{2} \)
23 \( 1 + 148 T + p^{3} T^{2} \)
29 \( 1 + 278 T + p^{3} T^{2} \)
31 \( 1 - 94 T + p^{3} T^{2} \)
37 \( 1 - 160 T + p^{3} T^{2} \)
41 \( 1 - 400 T + p^{3} T^{2} \)
43 \( 1 - 73 T + p^{3} T^{2} \)
47 \( 1 - 173 T + p^{3} T^{2} \)
53 \( 1 - 170 T + p^{3} T^{2} \)
59 \( 1 + 12 T + p^{3} T^{2} \)
61 \( 1 - 419 T + p^{3} T^{2} \)
67 \( 1 - 444 T + p^{3} T^{2} \)
71 \( 1 + 952 T + p^{3} T^{2} \)
73 \( 1 + 27 T + p^{3} T^{2} \)
79 \( 1 + 556 T + p^{3} T^{2} \)
83 \( 1 + 276 T + p^{3} T^{2} \)
89 \( 1 - 1386 T + p^{3} T^{2} \)
97 \( 1 - 130 T + p^{3} T^{2} \)
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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

−12.66524675210942946167990214791, −11.37074144325783268553232031785, −10.29191010019405655083642481763, −9.353394792963570512676483106449, −8.155930358516547018186436065885, −7.42952664568266259469790024665, −5.94890438424727878278728793423, −3.92289523343549717394309245019, −2.45054038779187925202972429966, 0, 2.45054038779187925202972429966, 3.92289523343549717394309245019, 5.94890438424727878278728793423, 7.42952664568266259469790024665, 8.155930358516547018186436065885, 9.353394792963570512676483106449, 10.29191010019405655083642481763, 11.37074144325783268553232031785, 12.66524675210942946167990214791

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