Properties

Label 2-1407-1407.185-c0-0-0
Degree $2$
Conductor $1407$
Sign $0.770 + 0.637i$
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  + (0.995 + 0.0950i)3-s + (0.786 − 0.618i)4-s + (−0.654 − 0.755i)7-s + (0.981 + 0.189i)9-s + (0.841 − 0.540i)12-s + (−1.16 − 0.600i)13-s + (0.235 − 0.971i)16-s + (−0.0359 − 0.186i)19-s + (−0.580 − 0.814i)21-s + (0.0475 + 0.998i)25-s + (0.959 + 0.281i)27-s + (−0.981 − 0.189i)28-s + (−0.0913 + 1.91i)31-s + (0.888 − 0.458i)36-s + (−0.0475 + 0.0824i)37-s + ⋯
L(s)  = 1  + (0.995 + 0.0950i)3-s + (0.786 − 0.618i)4-s + (−0.654 − 0.755i)7-s + (0.981 + 0.189i)9-s + (0.841 − 0.540i)12-s + (−1.16 − 0.600i)13-s + (0.235 − 0.971i)16-s + (−0.0359 − 0.186i)19-s + (−0.580 − 0.814i)21-s + (0.0475 + 0.998i)25-s + (0.959 + 0.281i)27-s + (−0.981 − 0.189i)28-s + (−0.0913 + 1.91i)31-s + (0.888 − 0.458i)36-s + (−0.0475 + 0.0824i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $0.770 + 0.637i$
Analytic conductor: \(0.702184\)
Root analytic conductor: \(0.837964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1407,\ (\ :0),\ 0.770 + 0.637i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.647879710\)
\(L(\frac12)\) \(\approx\) \(1.647879710\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.995 - 0.0950i)T \)
7 \( 1 + (0.654 + 0.755i)T \)
67 \( 1 + (0.841 - 0.540i)T \)
good2 \( 1 + (-0.786 + 0.618i)T^{2} \)
5 \( 1 + (-0.0475 - 0.998i)T^{2} \)
11 \( 1 + (0.0475 + 0.998i)T^{2} \)
13 \( 1 + (1.16 + 0.600i)T + (0.580 + 0.814i)T^{2} \)
17 \( 1 + (-0.959 - 0.281i)T^{2} \)
19 \( 1 + (0.0359 + 0.186i)T + (-0.928 + 0.371i)T^{2} \)
23 \( 1 + (-0.981 - 0.189i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.0913 - 1.91i)T + (-0.995 - 0.0950i)T^{2} \)
37 \( 1 + (0.0475 - 0.0824i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.235 - 0.971i)T^{2} \)
43 \( 1 + (-1.07 - 0.153i)T + (0.959 + 0.281i)T^{2} \)
47 \( 1 + (-0.654 + 0.755i)T^{2} \)
53 \( 1 + (0.723 + 0.690i)T^{2} \)
59 \( 1 + (0.580 - 0.814i)T^{2} \)
61 \( 1 + (1.44 - 1.37i)T + (0.0475 - 0.998i)T^{2} \)
71 \( 1 + (-0.723 - 0.690i)T^{2} \)
73 \( 1 + (-0.547 + 1.86i)T + (-0.841 - 0.540i)T^{2} \)
79 \( 1 + (-1.08 - 1.68i)T + (-0.415 + 0.909i)T^{2} \)
83 \( 1 + (0.0475 + 0.998i)T^{2} \)
89 \( 1 + (0.981 - 0.189i)T^{2} \)
97 \( 1 + (0.580 + 1.00i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.652768556992512494965159666223, −9.085067960272526365523432632529, −7.84609399978797304495257370204, −7.24126851972595454617822223075, −6.69770449239182493162363578505, −5.49300803877584302264964911186, −4.54534471068192349794495474681, −3.30879966185866007335839445360, −2.66720672491230161110063970927, −1.37546578361237762968400389448, 2.12307633189289024427545966301, 2.58306362419774008749270651687, 3.60288900835790033981760824491, 4.54328419358289065296078367147, 5.98956298133176529115311700651, 6.71537110816757238642503123265, 7.54899348788613932221918225185, 8.112251184285836074577967561584, 9.095769054319113263870577385156, 9.607460331281947776670963164834

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