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Properties

Label 2-1407-1407.305-c0-0-0
Degree $2$
Conductor $1407$
Sign $0.534 + 0.845i$
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 + (−0.5 − 0.866i)4-s + 7-s + (−0.499 − 0.866i)9-s + 0.999·12-s + (−1 − 1.73i)13-s + (−0.499 + 0.866i)16-s + (0.5 − 0.866i)19-s + (−0.5 + 0.866i)21-s + (−0.5 + 0.866i)25-s + 0.999·27-s + (−0.5 − 0.866i)28-s + (−1 − 1.73i)31-s + (−0.499 + 0.866i)36-s + (0.5 − 0.866i)37-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + 7-s + (−0.499 − 0.866i)9-s + 0.999·12-s + (−1 − 1.73i)13-s + (−0.499 + 0.866i)16-s + (0.5 − 0.866i)19-s + (−0.5 + 0.866i)21-s + (−0.5 + 0.866i)25-s + 0.999·27-s + (−0.5 − 0.866i)28-s + (−1 − 1.73i)31-s + (−0.499 + 0.866i)36-s + (0.5 − 0.866i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.534 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.534 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $0.534 + 0.845i$
Analytic conductor: \(0.702184\)
Root analytic conductor: \(0.837964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1407,\ (\ :0),\ 0.534 + 0.845i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7649868445\)
\(L(\frac12)\) \(\approx\) \(0.7649868445\)
\(L(1)\) 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 \( 1 + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 - 2T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.564804065835530545434044220339, −9.207667009347694526354304572468, −8.043756096560041165088773997183, −7.29391710821043325484498319807, −5.78233931381800355777307129813, −5.48206221897176564016011421122, −4.74504674665809501871201816951, −3.87065536142520092049584086505, −2.45273769102271691789334997642, −0.71333275705828915560384854391, 1.59139457834576014859570828707, 2.61084739671348037894752131954, 4.11963344652339218421162370444, 4.78693540181321847401971777309, 5.70894280497454751734975897355, 6.93686414969023179464337171152, 7.39148225058485251678806992161, 8.189293543236525923935824337689, 8.852433834026775735899325926882, 9.779648655661643768087538806079

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