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Properties

Label 2-3e2-3.2-c14-0-0
Degree $2$
Conductor $9$
Sign $-0.577 + 0.816i$
Analytic cond. $11.1896$
Root an. cond. $3.34508$
Motivic weight $14$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 148. i·2-s − 5.78e3·4-s + 4.16e4i·5-s − 1.41e6·7-s + 1.57e6i·8-s − 6.20e6·10-s − 2.60e7i·11-s + 1.08e7·13-s − 2.10e8i·14-s − 3.29e8·16-s − 1.18e8i·17-s − 1.32e9·19-s − 2.41e8i·20-s + 3.88e9·22-s + 2.14e9i·23-s + ⋯
L(s)  = 1  + 1.16i·2-s − 0.353·4-s + 0.533i·5-s − 1.71·7-s + 0.752i·8-s − 0.620·10-s − 1.33i·11-s + 0.173·13-s − 2.00i·14-s − 1.22·16-s − 0.288i·17-s − 1.48·19-s − 0.188i·20-s + 1.55·22-s + 0.630i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9\)    =    \(3^{2}\)
Sign: $-0.577 + 0.816i$
Analytic conductor: \(11.1896\)
Root analytic conductor: \(3.34508\)
Motivic weight: \(14\)
Rational: no
Arithmetic: yes
Character: $\chi_{9} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 9,\ (\ :7),\ -0.577 + 0.816i)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(0.245107 - 0.473510i\)
\(L(\frac12)\) \(\approx\) \(0.245107 - 0.473510i\)
\(L(8)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 - 148. iT - 1.63e4T^{2} \)
5 \( 1 - 4.16e4iT - 6.10e9T^{2} \)
7 \( 1 + 1.41e6T + 6.78e11T^{2} \)
11 \( 1 + 2.60e7iT - 3.79e14T^{2} \)
13 \( 1 - 1.08e7T + 3.93e15T^{2} \)
17 \( 1 + 1.18e8iT - 1.68e17T^{2} \)
19 \( 1 + 1.32e9T + 7.99e17T^{2} \)
23 \( 1 - 2.14e9iT - 1.15e19T^{2} \)
29 \( 1 - 2.83e10iT - 2.97e20T^{2} \)
31 \( 1 + 4.08e10T + 7.56e20T^{2} \)
37 \( 1 + 5.97e8T + 9.01e21T^{2} \)
41 \( 1 - 1.98e11iT - 3.79e22T^{2} \)
43 \( 1 - 1.49e10T + 7.38e22T^{2} \)
47 \( 1 - 4.13e11iT - 2.56e23T^{2} \)
53 \( 1 + 9.34e11iT - 1.37e24T^{2} \)
59 \( 1 - 3.83e12iT - 6.19e24T^{2} \)
61 \( 1 + 1.74e12T + 9.87e24T^{2} \)
67 \( 1 - 4.76e12T + 3.67e25T^{2} \)
71 \( 1 - 3.08e12iT - 8.27e25T^{2} \)
73 \( 1 - 1.09e12T + 1.22e26T^{2} \)
79 \( 1 + 1.08e13T + 3.68e26T^{2} \)
83 \( 1 + 2.26e13iT - 7.36e26T^{2} \)
89 \( 1 + 7.22e13iT - 1.95e27T^{2} \)
97 \( 1 - 1.82e13T + 6.52e27T^{2} \)
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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

−18.55520244698967325034223150299, −16.69655476794394964451450489201, −16.00018099503484298358173187335, −14.57222771832322463426617116024, −13.07689393698274629521695496324, −10.86960122187777714101271133098, −8.867664651187508717400666620379, −6.93657072230974131490683619709, −5.94133025700389096806718444679, −3.13744306903475874467416412202, 0.22232332782510490043709591963, 2.23331339891074829312991735396, 3.98043664429537852953738095527, 6.64548297180009168093485211262, 9.315197610676521754612725895433, 10.44892660387659724830381927209, 12.43711172191605546496895633811, 12.95431426048440424565903153411, 15.44523323999200412915367626264, 16.82160423143325219832285798118

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