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Properties

Label 2-1407-1407.1262-c0-0-0
Degree $2$
Conductor $1407$
Sign $0.470 + 0.882i$
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  + (0.580 − 0.814i)3-s + (0.928 + 0.371i)4-s + (−0.654 − 0.755i)7-s + (−0.327 − 0.945i)9-s + (0.841 − 0.540i)12-s + (−0.0623 − 1.30i)13-s + (0.723 + 0.690i)16-s + (−0.379 + 1.09i)19-s + (−0.995 + 0.0950i)21-s + (−0.888 − 0.458i)25-s + (−0.959 − 0.281i)27-s + (−0.327 − 0.945i)28-s + (1.70 − 0.879i)31-s + (0.0475 − 0.998i)36-s + (0.888 + 1.53i)37-s + ⋯
L(s)  = 1  + (0.580 − 0.814i)3-s + (0.928 + 0.371i)4-s + (−0.654 − 0.755i)7-s + (−0.327 − 0.945i)9-s + (0.841 − 0.540i)12-s + (−0.0623 − 1.30i)13-s + (0.723 + 0.690i)16-s + (−0.379 + 1.09i)19-s + (−0.995 + 0.0950i)21-s + (−0.888 − 0.458i)25-s + (−0.959 − 0.281i)27-s + (−0.327 − 0.945i)28-s + (1.70 − 0.879i)31-s + (0.0475 − 0.998i)36-s + (0.888 + 1.53i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.470 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.470 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $0.470 + 0.882i$
Analytic conductor: \(0.702184\)
Root analytic conductor: \(0.837964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (1262, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1407,\ (\ :0),\ 0.470 + 0.882i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.476999024\)
\(L(\frac12)\) \(\approx\) \(1.476999024\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.580 + 0.814i)T \)
7 \( 1 + (0.654 + 0.755i)T \)
67 \( 1 + (-0.841 + 0.540i)T \)
good2 \( 1 + (-0.928 - 0.371i)T^{2} \)
5 \( 1 + (0.888 + 0.458i)T^{2} \)
11 \( 1 + (0.888 + 0.458i)T^{2} \)
13 \( 1 + (0.0623 + 1.30i)T + (-0.995 + 0.0950i)T^{2} \)
17 \( 1 + (0.959 + 0.281i)T^{2} \)
19 \( 1 + (0.379 - 1.09i)T + (-0.786 - 0.618i)T^{2} \)
23 \( 1 + (0.327 + 0.945i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-1.70 + 0.879i)T + (0.580 - 0.814i)T^{2} \)
37 \( 1 + (-0.888 - 1.53i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.723 + 0.690i)T^{2} \)
43 \( 1 + (0.239 - 1.66i)T + (-0.959 - 0.281i)T^{2} \)
47 \( 1 + (0.654 - 0.755i)T^{2} \)
53 \( 1 + (-0.235 + 0.971i)T^{2} \)
59 \( 1 + (0.995 + 0.0950i)T^{2} \)
61 \( 1 + (-0.273 - 1.12i)T + (-0.888 + 0.458i)T^{2} \)
71 \( 1 + (-0.235 + 0.971i)T^{2} \)
73 \( 1 + (1.38 + 0.407i)T + (0.841 + 0.540i)T^{2} \)
79 \( 1 + (1.49 - 0.961i)T + (0.415 - 0.909i)T^{2} \)
83 \( 1 + (0.888 + 0.458i)T^{2} \)
89 \( 1 + (0.327 - 0.945i)T^{2} \)
97 \( 1 + (-0.995 + 1.72i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.962907446335165770987222005750, −8.378985528282451426213095527259, −7.944283262874350797579584684226, −7.34467327301310360130824657751, −6.28003335222400526487066192879, −6.04947016378450773594107099282, −4.22802609178074866917547628699, −3.22666225271534104663883951505, −2.60307363163198700874365963746, −1.22566663186371219611636998664, 2.04893334518211494253692157137, 2.71104064165307744791910123975, 3.79021521314199557439804859324, 4.85234337893673949695116151051, 5.78699346790075643812006246555, 6.62308254427170782411340762501, 7.37720189096458398439789213413, 8.538024132363717445251656635292, 9.173537798975548997982005621960, 9.823018430143993360731210045468

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