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Properties

Label 2-1407-1407.473-c0-0-0
Degree $2$
Conductor $1407$
Sign $0.996 - 0.0815i$
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  + (−0.888 + 0.458i)3-s + (−0.654 + 0.755i)4-s + (0.723 − 0.690i)7-s + (0.580 − 0.814i)9-s + (0.235 − 0.971i)12-s + (0.195 − 0.807i)13-s + (−0.142 − 0.989i)16-s + (0.0395 − 0.0865i)19-s + (−0.327 + 0.945i)21-s + (0.235 − 0.971i)25-s + (−0.142 + 0.989i)27-s + (0.0475 + 0.998i)28-s + (0.273 + 0.0801i)31-s + (0.235 + 0.971i)36-s + 1.44·37-s + ⋯
L(s)  = 1  + (−0.888 + 0.458i)3-s + (−0.654 + 0.755i)4-s + (0.723 − 0.690i)7-s + (0.580 − 0.814i)9-s + (0.235 − 0.971i)12-s + (0.195 − 0.807i)13-s + (−0.142 − 0.989i)16-s + (0.0395 − 0.0865i)19-s + (−0.327 + 0.945i)21-s + (0.235 − 0.971i)25-s + (−0.142 + 0.989i)27-s + (0.0475 + 0.998i)28-s + (0.273 + 0.0801i)31-s + (0.235 + 0.971i)36-s + 1.44·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $0.996 - 0.0815i$
Analytic conductor: \(0.702184\)
Root analytic conductor: \(0.837964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (473, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1407,\ (\ :0),\ 0.996 - 0.0815i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7557065039\)
\(L(\frac12)\) \(\approx\) \(0.7557065039\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.888 - 0.458i)T \)
7 \( 1 + (-0.723 + 0.690i)T \)
67 \( 1 + (-0.723 - 0.690i)T \)
good2 \( 1 + (0.654 - 0.755i)T^{2} \)
5 \( 1 + (-0.235 + 0.971i)T^{2} \)
11 \( 1 + (0.959 - 0.281i)T^{2} \)
13 \( 1 + (-0.195 + 0.807i)T + (-0.888 - 0.458i)T^{2} \)
17 \( 1 + (0.786 + 0.618i)T^{2} \)
19 \( 1 + (-0.0395 + 0.0865i)T + (-0.654 - 0.755i)T^{2} \)
23 \( 1 + (-0.415 - 0.909i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.273 - 0.0801i)T + (0.841 + 0.540i)T^{2} \)
37 \( 1 - 1.44T + T^{2} \)
41 \( 1 + (0.786 + 0.618i)T^{2} \)
43 \( 1 + (-1.25 - 1.45i)T + (-0.142 + 0.989i)T^{2} \)
47 \( 1 + (-0.580 + 0.814i)T^{2} \)
53 \( 1 + (0.786 - 0.618i)T^{2} \)
59 \( 1 + (-0.0475 + 0.998i)T^{2} \)
61 \( 1 + (0.0135 - 0.0941i)T + (-0.959 - 0.281i)T^{2} \)
71 \( 1 + (-0.928 - 0.371i)T^{2} \)
73 \( 1 + (1.45 - 0.584i)T + (0.723 - 0.690i)T^{2} \)
79 \( 1 + (-0.341 + 1.40i)T + (-0.888 - 0.458i)T^{2} \)
83 \( 1 + (-0.723 - 0.690i)T^{2} \)
89 \( 1 + (-0.580 - 0.814i)T^{2} \)
97 \( 1 + (-0.888 + 1.53i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.876636001100153840216293999658, −8.990828448084062255386174223354, −8.066205727795276259335641272535, −7.49454705466183824306186356181, −6.42977956780760558997953559244, −5.45416372556766821309084852790, −4.54860194427684752054646913337, −4.09411968517431244023374233363, −2.91195202700681442229945510271, −0.899999035950006406796953055119, 1.20702446178463269520350146867, 2.21957029936536889199148648764, 4.09480001820336509458926586223, 4.89432356900245983968083639100, 5.60405538950547791239178732372, 6.24115001083918045503915682836, 7.22348740328965564644721802044, 8.165901188827165617544076870379, 9.032094341510073260332286169993, 9.686858063466963154315519305391

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