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Properties

Label 2-1407-1407.317-c0-0-0
Degree $2$
Conductor $1407$
Sign $0.706 - 0.708i$
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 + (−0.654 + 0.755i)4-s + (0.235 + 0.971i)7-s + (−0.995 − 0.0950i)9-s + (0.723 + 0.690i)12-s + (0.601 + 0.573i)13-s + (−0.142 − 0.989i)16-s + (−0.738 + 1.61i)19-s + (0.981 − 0.189i)21-s + (0.723 + 0.690i)25-s + (−0.142 + 0.989i)27-s + (−0.888 − 0.458i)28-s + (0.273 + 0.0801i)31-s + (0.723 − 0.690i)36-s + 0.471·37-s + ⋯
L(s)  = 1  + (0.0475 − 0.998i)3-s + (−0.654 + 0.755i)4-s + (0.235 + 0.971i)7-s + (−0.995 − 0.0950i)9-s + (0.723 + 0.690i)12-s + (0.601 + 0.573i)13-s + (−0.142 − 0.989i)16-s + (−0.738 + 1.61i)19-s + (0.981 − 0.189i)21-s + (0.723 + 0.690i)25-s + (−0.142 + 0.989i)27-s + (−0.888 − 0.458i)28-s + (0.273 + 0.0801i)31-s + (0.723 − 0.690i)36-s + 0.471·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.706 - 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.706 - 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $0.706 - 0.708i$
Analytic conductor: \(0.702184\)
Root analytic conductor: \(0.837964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1407,\ (\ :0),\ 0.706 - 0.708i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8799829574\)
\(L(\frac12)\) \(\approx\) \(0.8799829574\)
\(L(1)\) 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 \( 1 + (-0.235 - 0.971i)T \)
67 \( 1 + (-0.235 + 0.971i)T \)
good2 \( 1 + (0.654 - 0.755i)T^{2} \)
5 \( 1 + (-0.723 - 0.690i)T^{2} \)
11 \( 1 + (0.959 - 0.281i)T^{2} \)
13 \( 1 + (-0.601 - 0.573i)T + (0.0475 + 0.998i)T^{2} \)
17 \( 1 + (-0.928 + 0.371i)T^{2} \)
19 \( 1 + (0.738 - 1.61i)T + (-0.654 - 0.755i)T^{2} \)
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 \( 1 - 0.471T + T^{2} \)
41 \( 1 + (-0.928 + 0.371i)T^{2} \)
43 \( 1 + (-1.25 - 1.45i)T + (-0.142 + 0.989i)T^{2} \)
47 \( 1 + (0.995 + 0.0950i)T^{2} \)
53 \( 1 + (-0.928 - 0.371i)T^{2} \)
59 \( 1 + (0.888 - 0.458i)T^{2} \)
61 \( 1 + (-0.252 + 1.75i)T + (-0.959 - 0.281i)T^{2} \)
71 \( 1 + (0.786 - 0.618i)T^{2} \)
73 \( 1 + (1.45 + 1.14i)T + (0.235 + 0.971i)T^{2} \)
79 \( 1 + (-0.341 - 0.325i)T + (0.0475 + 0.998i)T^{2} \)
83 \( 1 + (-0.235 + 0.971i)T^{2} \)
89 \( 1 + (0.995 - 0.0950i)T^{2} \)
97 \( 1 + (0.0475 + 0.0824i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.467269842717169346454813421908, −8.903402598615596080364379394684, −8.157777420602681767184277555745, −7.72205385520305486193607404762, −6.54185678394448796375980573281, −5.88534046677350688486029879031, −4.85738095195580844886064367039, −3.72138331792197853799744341968, −2.73189526782186096002377238690, −1.58705271891404224261107386708, 0.789305932020606694103965802392, 2.65359040985140493965634857432, 3.97765434630225987458361541324, 4.47533463981757084797213883066, 5.31047383581860312223219281495, 6.18090641834153463018637601963, 7.19005554283685262820096334754, 8.451227052905097668451281167257, 8.841784835373826744705345538316, 9.755781322050503076575268351953

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