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Properties

Label 2-1407-1407.1241-c0-0-0
Degree $2$
Conductor $1407$
Sign $-0.679 + 0.733i$
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  + (−0.786 + 0.618i)3-s + (0.0475 + 0.998i)4-s + (−0.959 − 0.281i)7-s + (0.235 − 0.971i)9-s + (−0.654 − 0.755i)12-s + (−1.88 + 0.363i)13-s + (−0.995 + 0.0950i)16-s + (−0.370 − 1.52i)19-s + (0.928 − 0.371i)21-s + (−0.327 + 0.945i)25-s + (0.415 + 0.909i)27-s + (0.235 − 0.971i)28-s + (−0.271 − 0.785i)31-s + (0.981 + 0.189i)36-s + (0.327 + 0.566i)37-s + ⋯
L(s)  = 1  + (−0.786 + 0.618i)3-s + (0.0475 + 0.998i)4-s + (−0.959 − 0.281i)7-s + (0.235 − 0.971i)9-s + (−0.654 − 0.755i)12-s + (−1.88 + 0.363i)13-s + (−0.995 + 0.0950i)16-s + (−0.370 − 1.52i)19-s + (0.928 − 0.371i)21-s + (−0.327 + 0.945i)25-s + (0.415 + 0.909i)27-s + (0.235 − 0.971i)28-s + (−0.271 − 0.785i)31-s + (0.981 + 0.189i)36-s + (0.327 + 0.566i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.679 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.679 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $-0.679 + 0.733i$
Analytic conductor: \(0.702184\)
Root analytic conductor: \(0.837964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (1241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1407,\ (\ :0),\ -0.679 + 0.733i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05793230627\)
\(L(\frac12)\) \(\approx\) \(0.05793230627\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.786 - 0.618i)T \)
7 \( 1 + (0.959 + 0.281i)T \)
67 \( 1 + (0.654 + 0.755i)T \)
good2 \( 1 + (-0.0475 - 0.998i)T^{2} \)
5 \( 1 + (0.327 - 0.945i)T^{2} \)
11 \( 1 + (0.327 - 0.945i)T^{2} \)
13 \( 1 + (1.88 - 0.363i)T + (0.928 - 0.371i)T^{2} \)
17 \( 1 + (-0.415 - 0.909i)T^{2} \)
19 \( 1 + (0.370 + 1.52i)T + (-0.888 + 0.458i)T^{2} \)
23 \( 1 + (-0.235 + 0.971i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.271 + 0.785i)T + (-0.786 + 0.618i)T^{2} \)
37 \( 1 + (-0.327 - 0.566i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.995 + 0.0950i)T^{2} \)
43 \( 1 + (1.10 + 0.708i)T + (0.415 + 0.909i)T^{2} \)
47 \( 1 + (0.959 - 0.281i)T^{2} \)
53 \( 1 + (-0.580 - 0.814i)T^{2} \)
59 \( 1 + (-0.928 - 0.371i)T^{2} \)
61 \( 1 + (0.911 - 1.28i)T + (-0.327 - 0.945i)T^{2} \)
71 \( 1 + (-0.580 - 0.814i)T^{2} \)
73 \( 1 + (0.827 + 1.81i)T + (-0.654 + 0.755i)T^{2} \)
79 \( 1 + (-0.428 - 0.494i)T + (-0.142 + 0.989i)T^{2} \)
83 \( 1 + (0.327 - 0.945i)T^{2} \)
89 \( 1 + (-0.235 - 0.971i)T^{2} \)
97 \( 1 + (0.928 - 1.60i)T + (-0.5 - 0.866i)T^{2} \)
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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

−10.18169710104413668388227079303, −9.417468046560342178534326526576, −8.992347757772745744445235043228, −7.55781995661731889415978838258, −7.04673471522345170551958860737, −6.30798565864642910444440703818, −5.05659399170289422330321655533, −4.40581578617761646452125832669, −3.43494498577970536908542986436, −2.50719680252722461543408210229, 0.04857812477881242410261281374, 1.74351475039054145304584811431, 2.75494743148828987696885221818, 4.39074607023156428019680426648, 5.32164219689297595861833521931, 5.93472969374076752580437576378, 6.66027342369657994228943897155, 7.38624051709389557759051251079, 8.362455382099722748840636287377, 9.644993243633847174377207147195

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