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Properties

Label 2-1407-1407.878-c0-0-0
Degree $2$
Conductor $1407$
Sign $0.352 + 0.935i$
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  + (0.723 − 0.690i)3-s + (−0.415 − 0.909i)4-s + (0.786 + 0.618i)7-s + (0.0475 − 0.998i)9-s + (−0.928 − 0.371i)12-s + (1.56 + 0.625i)13-s + (−0.654 + 0.755i)16-s + (−1.05 − 1.63i)19-s + (0.995 − 0.0950i)21-s + (0.928 + 0.371i)25-s + (−0.654 − 0.755i)27-s + (0.235 − 0.971i)28-s + (−0.186 + 1.29i)31-s + (−0.928 + 0.371i)36-s − 1.57·37-s + ⋯
L(s)  = 1  + (0.723 − 0.690i)3-s + (−0.415 − 0.909i)4-s + (0.786 + 0.618i)7-s + (0.0475 − 0.998i)9-s + (−0.928 − 0.371i)12-s + (1.56 + 0.625i)13-s + (−0.654 + 0.755i)16-s + (−1.05 − 1.63i)19-s + (0.995 − 0.0950i)21-s + (0.928 + 0.371i)25-s + (−0.654 − 0.755i)27-s + (0.235 − 0.971i)28-s + (−0.186 + 1.29i)31-s + (−0.928 + 0.371i)36-s − 1.57·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.352 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.352 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $0.352 + 0.935i$
Analytic conductor: \(0.702184\)
Root analytic conductor: \(0.837964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (878, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1407,\ (\ :0),\ 0.352 + 0.935i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.389644172\)
\(L(\frac12)\) \(\approx\) \(1.389644172\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.723 + 0.690i)T \)
7 \( 1 + (-0.786 - 0.618i)T \)
67 \( 1 + (-0.786 + 0.618i)T \)
good2 \( 1 + (0.415 + 0.909i)T^{2} \)
5 \( 1 + (-0.928 - 0.371i)T^{2} \)
11 \( 1 + (-0.142 - 0.989i)T^{2} \)
13 \( 1 + (-1.56 - 0.625i)T + (0.723 + 0.690i)T^{2} \)
17 \( 1 + (0.981 - 0.189i)T^{2} \)
19 \( 1 + (1.05 + 1.63i)T + (-0.415 + 0.909i)T^{2} \)
23 \( 1 + (-0.841 - 0.540i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.186 - 1.29i)T + (-0.959 - 0.281i)T^{2} \)
37 \( 1 + 1.57T + T^{2} \)
41 \( 1 + (-0.981 + 0.189i)T^{2} \)
43 \( 1 + (1.80 + 0.822i)T + (0.654 + 0.755i)T^{2} \)
47 \( 1 + (0.0475 - 0.998i)T^{2} \)
53 \( 1 + (0.981 + 0.189i)T^{2} \)
59 \( 1 + (0.235 + 0.971i)T^{2} \)
61 \( 1 + (-0.308 - 0.356i)T + (-0.142 + 0.989i)T^{2} \)
71 \( 1 + (0.327 + 0.945i)T^{2} \)
73 \( 1 + (0.357 + 0.123i)T + (0.786 + 0.618i)T^{2} \)
79 \( 1 + (0.459 - 1.14i)T + (-0.723 - 0.690i)T^{2} \)
83 \( 1 + (-0.786 + 0.618i)T^{2} \)
89 \( 1 + (0.0475 + 0.998i)T^{2} \)
97 \( 1 + (-0.723 + 1.25i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.199974696379508621438608680624, −8.684207990058669985136876214023, −8.464414084503378272879218197559, −6.92123739826765092698661125546, −6.53803564811638215340342083451, −5.42566815102587937939382437287, −4.61244701195931183776341475508, −3.45554574573497614742787305281, −2.11980221493369283044055117085, −1.30306125832757166067418112998, 1.78571088385958187849219099854, 3.23422849508763197226332084526, 3.86221495681054731216975404652, 4.53038950722479049565732134839, 5.58457109572002875769686189691, 6.83470221294815143290054531532, 8.024148770878127749492447494273, 8.212582678604228038430327326657, 8.820347895059388768783105124733, 9.935638261809347732738169012742

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