Properties

Label 2-294-1.1-c1-0-0
Degree $2$
Conductor $294$
Sign $1$
Analytic cond. $2.34760$
Root an. cond. $1.53218$
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 − 3-s + 4-s − 3·5-s + 6-s − 8-s + 9-s + 3·10-s + 3·11-s − 12-s + 4·13-s + 3·15-s + 16-s − 18-s + 4·19-s − 3·20-s − 3·22-s + 24-s + 4·25-s − 4·26-s − 27-s + 9·29-s − 3·30-s + 31-s − 32-s − 3·33-s + 36-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.948·10-s + 0.904·11-s − 0.288·12-s + 1.10·13-s + 0.774·15-s + 1/4·16-s − 0.235·18-s + 0.917·19-s − 0.670·20-s − 0.639·22-s + 0.204·24-s + 4/5·25-s − 0.784·26-s − 0.192·27-s + 1.67·29-s − 0.547·30-s + 0.179·31-s − 0.176·32-s − 0.522·33-s + 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(2.34760\)
Root analytic conductor: \(1.53218\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6693039983\)
\(L(\frac12)\) \(\approx\) \(0.6693039983\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 \)
good5 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−11.68014818696489555780067605703, −11.01449612120722930985164678406, −9.959946615270396196728666018490, −8.804654554318459390164507311566, −8.000167698268580301677238135628, −7.00728426330403345958805784350, −6.07864651997567107362244436260, −4.49123641406362856214796748304, −3.36779260852424710508563371535, −1.01560111630863456380812991130, 1.01560111630863456380812991130, 3.36779260852424710508563371535, 4.49123641406362856214796748304, 6.07864651997567107362244436260, 7.00728426330403345958805784350, 8.000167698268580301677238135628, 8.804654554318459390164507311566, 9.959946615270396196728666018490, 11.01449612120722930985164678406, 11.68014818696489555780067605703

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