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Properties

Label 2-1407-1407.1406-c0-0-3
Degree $2$
Conductor $1407$
Sign $-0.5 - 0.866i$
Analytic cond. $0.702184$
Root an. cond. $0.837964$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)3-s − 4-s + 1.73i·5-s + 7-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)12-s + 13-s + (−1.49 + 0.866i)15-s + 16-s − 1.73i·20-s + (0.5 + 0.866i)21-s − 1.73i·23-s − 1.99·25-s − 0.999·27-s − 28-s + 1.73i·29-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s − 4-s + 1.73i·5-s + 7-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)12-s + 13-s + (−1.49 + 0.866i)15-s + 16-s − 1.73i·20-s + (0.5 + 0.866i)21-s − 1.73i·23-s − 1.99·25-s − 0.999·27-s − 28-s + 1.73i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $-0.5 - 0.866i$
Analytic conductor: \(0.702184\)
Root analytic conductor: \(0.837964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (1406, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1407,\ (\ :0),\ -0.5 - 0.866i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.123081839\)
\(L(\frac12)\) \(\approx\) \(1.123081839\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 - T \)
67 \( 1 + T \)
good2 \( 1 + T^{2} \)
5 \( 1 - 1.73iT - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - T + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + 1.73iT - T^{2} \)
29 \( 1 - 1.73iT - T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + 1.73iT - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - 2T + T^{2} \)
71 \( 1 + 1.73iT - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - 2T + T^{2} \)
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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

−10.31454859500329043649560120305, −9.020283489823640449024271645797, −8.645513078107519362349667030793, −7.75727478615204530295961718680, −6.87137438498741336914144040644, −5.71615054205501623598027078334, −4.88828239533169303829711980575, −3.86674988596198155598335469439, −3.32296513914252563207202622502, −2.09719487096067002268432789659, 1.01771996531169126847992892333, 1.76036008249192112288343301193, 3.62522462780689819407713846962, 4.37772399079527007277173375412, 5.35447971944435143095484741255, 5.87994107162348246500486711202, 7.45049664580676613416950803660, 8.184150452737844542227076653834, 8.518715970110043886587835706593, 9.212736921882077230823285567473

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