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Properties

Label 2-1407-1407.614-c0-0-0
Degree $2$
Conductor $1407$
Sign $0.747 - 0.663i$
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 + (1.53 + 0.532i)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 + (1.53 + 0.532i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.747 - 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.747 - 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $0.747 - 0.663i$
Analytic conductor: \(0.702184\)
Root analytic conductor: \(0.837964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (614, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1407,\ (\ :0),\ 0.747 - 0.663i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6420799442\)
\(L(\frac12)\) \(\approx\) \(0.6420799442\)
\(L(1)\) 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 \( 1 + (-1.53 - 0.532i)T + (0.786 + 0.618i)T^{2} \)
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 \( 1 + (-1.07 - 0.153i)T + (0.959 + 0.281i)T^{2} \)
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 \( 1 + (0.388 - 1.32i)T + (-0.841 - 0.540i)T^{2} \)
79 \( 1 + (0.495 + 0.770i)T + (-0.415 + 0.909i)T^{2} \)
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

−9.901856726297097184438791561357, −9.385231638955448127785286143506, −8.456078958716508701647500173433, −7.37506145836669980284426500343, −6.28265545178594652400037003435, −5.75661374863322279247320212411, −4.47940797742722204216111463041, −4.24209904641081236689805199366, −3.14093269033519140395377492348, −1.04075624889804836612365351607, 0.791816411765763963408926329982, 2.61150516759520578097839292180, 3.44760181637041766633349987890, 4.88410736663722131689612762352, 5.53844213419906564155465934518, 6.21715899938021674867021045876, 7.44495030880057119435210489176, 7.898095936012907115548188884160, 8.844199907365650518942476227671, 9.572231506847630362631757295562

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