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Properties

Label 2-1407-1407.404-c0-0-0
Degree $2$
Conductor $1407$
Sign $-0.673 - 0.739i$
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  + (0.235 + 0.971i)3-s + (−0.415 − 0.909i)4-s + (−0.928 + 0.371i)7-s + (−0.888 + 0.458i)9-s + (0.786 − 0.618i)12-s + (−1.32 + 1.04i)13-s + (−0.654 + 0.755i)16-s + (0.746 + 1.16i)19-s + (−0.580 − 0.814i)21-s + (−0.786 + 0.618i)25-s + (−0.654 − 0.755i)27-s + (0.723 + 0.690i)28-s + (−0.186 + 1.29i)31-s + (0.786 + 0.618i)36-s + 1.85·37-s + ⋯
L(s)  = 1  + (0.235 + 0.971i)3-s + (−0.415 − 0.909i)4-s + (−0.928 + 0.371i)7-s + (−0.888 + 0.458i)9-s + (0.786 − 0.618i)12-s + (−1.32 + 1.04i)13-s + (−0.654 + 0.755i)16-s + (0.746 + 1.16i)19-s + (−0.580 − 0.814i)21-s + (−0.786 + 0.618i)25-s + (−0.654 − 0.755i)27-s + (0.723 + 0.690i)28-s + (−0.186 + 1.29i)31-s + (0.786 + 0.618i)36-s + 1.85·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.673 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.673 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $-0.673 - 0.739i$
Analytic conductor: \(0.702184\)
Root analytic conductor: \(0.837964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (404, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1407,\ (\ :0),\ -0.673 - 0.739i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5782201490\)
\(L(\frac12)\) \(\approx\) \(0.5782201490\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.235 - 0.971i)T \)
7 \( 1 + (0.928 - 0.371i)T \)
67 \( 1 + (0.928 + 0.371i)T \)
good2 \( 1 + (0.415 + 0.909i)T^{2} \)
5 \( 1 + (0.786 - 0.618i)T^{2} \)
11 \( 1 + (-0.142 - 0.989i)T^{2} \)
13 \( 1 + (1.32 - 1.04i)T + (0.235 - 0.971i)T^{2} \)
17 \( 1 + (-0.327 + 0.945i)T^{2} \)
19 \( 1 + (-0.746 - 1.16i)T + (-0.415 + 0.909i)T^{2} \)
23 \( 1 + (-0.841 - 0.540i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.186 - 1.29i)T + (-0.959 - 0.281i)T^{2} \)
37 \( 1 - 1.85T + T^{2} \)
41 \( 1 + (0.327 - 0.945i)T^{2} \)
43 \( 1 + (1.80 + 0.822i)T + (0.654 + 0.755i)T^{2} \)
47 \( 1 + (-0.888 + 0.458i)T^{2} \)
53 \( 1 + (-0.327 - 0.945i)T^{2} \)
59 \( 1 + (0.723 - 0.690i)T^{2} \)
61 \( 1 + (-0.947 - 1.09i)T + (-0.142 + 0.989i)T^{2} \)
71 \( 1 + (-0.981 - 0.189i)T^{2} \)
73 \( 1 + (0.357 + 1.85i)T + (-0.928 + 0.371i)T^{2} \)
79 \( 1 + (0.459 + 0.584i)T + (-0.235 + 0.971i)T^{2} \)
83 \( 1 + (0.928 + 0.371i)T^{2} \)
89 \( 1 + (-0.888 - 0.458i)T^{2} \)
97 \( 1 + (-0.235 - 0.408i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.982317149934772924092566579894, −9.366746412719987506261536360998, −8.885615985024382457858975395658, −7.70514677940321892407559905704, −6.61484348165345408715439146369, −5.72694736173881162336802181072, −5.06028313583295112784666751871, −4.17004759072200348752711109511, −3.19743107239729364482923068249, −1.96311762615408353584396952253, 0.44008669005818252623911333446, 2.55917015063685397446155118957, 3.04832100792922303820175144633, 4.19937472803547417993388808974, 5.33781076368097200159509849882, 6.40054653459018322017641236468, 7.25760834106067355669396579817, 7.72268596420200219255591764321, 8.441894505629287252191373827512, 9.581410739724046786982603981988

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