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Properties

Label 2-1407-1407.1256-c0-0-0
Degree $2$
Conductor $1407$
Sign $-0.336 + 0.941i$
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  + (−0.0475 − 0.998i)3-s + (0.327 − 0.945i)4-s + (0.415 + 0.909i)7-s + (−0.995 + 0.0950i)9-s + (−0.959 − 0.281i)12-s + (−0.195 − 0.807i)13-s + (−0.786 − 0.618i)16-s + (0.189 − 1.98i)19-s + (0.888 − 0.458i)21-s + (0.723 − 0.690i)25-s + (0.142 + 0.989i)27-s + (0.995 − 0.0950i)28-s + (−0.205 − 0.196i)31-s + (−0.235 + 0.971i)36-s + (−0.723 + 1.25i)37-s + ⋯
L(s)  = 1  + (−0.0475 − 0.998i)3-s + (0.327 − 0.945i)4-s + (0.415 + 0.909i)7-s + (−0.995 + 0.0950i)9-s + (−0.959 − 0.281i)12-s + (−0.195 − 0.807i)13-s + (−0.786 − 0.618i)16-s + (0.189 − 1.98i)19-s + (0.888 − 0.458i)21-s + (0.723 − 0.690i)25-s + (0.142 + 0.989i)27-s + (0.995 − 0.0950i)28-s + (−0.205 − 0.196i)31-s + (−0.235 + 0.971i)36-s + (−0.723 + 1.25i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $-0.336 + 0.941i$
Analytic conductor: \(0.702184\)
Root analytic conductor: \(0.837964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (1256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1407,\ (\ :0),\ -0.336 + 0.941i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.106568769\)
\(L(\frac12)\) \(\approx\) \(1.106568769\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.0475 + 0.998i)T \)
7 \( 1 + (-0.415 - 0.909i)T \)
67 \( 1 + (-0.959 - 0.281i)T \)
good2 \( 1 + (-0.327 + 0.945i)T^{2} \)
5 \( 1 + (-0.723 + 0.690i)T^{2} \)
11 \( 1 + (0.723 - 0.690i)T^{2} \)
13 \( 1 + (0.195 + 0.807i)T + (-0.888 + 0.458i)T^{2} \)
17 \( 1 + (-0.142 - 0.989i)T^{2} \)
19 \( 1 + (-0.189 + 1.98i)T + (-0.981 - 0.189i)T^{2} \)
23 \( 1 + (0.995 - 0.0950i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.205 + 0.196i)T + (0.0475 + 0.998i)T^{2} \)
37 \( 1 + (0.723 - 1.25i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.786 - 0.618i)T^{2} \)
43 \( 1 + (-0.425 - 0.368i)T + (0.142 + 0.989i)T^{2} \)
47 \( 1 + (0.415 - 0.909i)T^{2} \)
53 \( 1 + (0.928 - 0.371i)T^{2} \)
59 \( 1 + (-0.888 - 0.458i)T^{2} \)
61 \( 1 + (-0.0883 - 0.0353i)T + (0.723 + 0.690i)T^{2} \)
71 \( 1 + (-0.928 + 0.371i)T^{2} \)
73 \( 1 + (-1.22 + 0.175i)T + (0.959 - 0.281i)T^{2} \)
79 \( 1 + (0.388 - 1.32i)T + (-0.841 - 0.540i)T^{2} \)
83 \( 1 + (0.723 - 0.690i)T^{2} \)
89 \( 1 + (-0.995 - 0.0950i)T^{2} \)
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

−9.366399026780395193693693815058, −8.700554066413266512734868289357, −7.86582464403587965879408527391, −6.89103077668569545862340335219, −6.34011715572310693632550363434, −5.34778348239569976926981530623, −4.90661875802464345734879396717, −2.88082074858647685223307476729, −2.26059645809018018219264242529, −0.948077764025661497540929577030, 1.93333754539138299617388778221, 3.39399157638198036425214688226, 3.89119462172122166592065521473, 4.73730989523265134215672634605, 5.79567320094437214225563744683, 6.90293754222054285939379048791, 7.63491192759919844553051351414, 8.413961305900817678323173881539, 9.173368121616480003720165141751, 10.07540418610243834328391294036

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