Properties

Label 2-1407-1407.1346-c0-0-0
Degree $2$
Conductor $1407$
Sign $-0.992 - 0.125i$
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  + (−0.995 + 0.0950i)3-s + (−0.142 + 0.989i)4-s + (−0.888 − 0.458i)7-s + (0.981 − 0.189i)9-s + (0.0475 − 0.998i)12-s + (−0.0623 + 1.30i)13-s + (−0.959 − 0.281i)16-s + (−0.759 + 0.876i)19-s + (0.928 + 0.371i)21-s + (0.0475 − 0.998i)25-s + (−0.959 + 0.281i)27-s + (0.580 − 0.814i)28-s + (−1.61 + 1.03i)31-s + (0.0475 + 0.998i)36-s − 1.77·37-s + ⋯
L(s)  = 1  + (−0.995 + 0.0950i)3-s + (−0.142 + 0.989i)4-s + (−0.888 − 0.458i)7-s + (0.981 − 0.189i)9-s + (0.0475 − 0.998i)12-s + (−0.0623 + 1.30i)13-s + (−0.959 − 0.281i)16-s + (−0.759 + 0.876i)19-s + (0.928 + 0.371i)21-s + (0.0475 − 0.998i)25-s + (−0.959 + 0.281i)27-s + (0.580 − 0.814i)28-s + (−1.61 + 1.03i)31-s + (0.0475 + 0.998i)36-s − 1.77·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $-0.992 - 0.125i$
Analytic conductor: \(0.702184\)
Root analytic conductor: \(0.837964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (1346, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1407,\ (\ :0),\ -0.992 - 0.125i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2702209745\)
\(L(\frac12)\) \(\approx\) \(0.2702209745\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.995 - 0.0950i)T \)
7 \( 1 + (0.888 + 0.458i)T \)
67 \( 1 + (0.888 - 0.458i)T \)
good2 \( 1 + (0.142 - 0.989i)T^{2} \)
5 \( 1 + (-0.0475 + 0.998i)T^{2} \)
11 \( 1 + (-0.841 - 0.540i)T^{2} \)
13 \( 1 + (0.0623 - 1.30i)T + (-0.995 - 0.0950i)T^{2} \)
17 \( 1 + (-0.723 - 0.690i)T^{2} \)
19 \( 1 + (0.759 - 0.876i)T + (-0.142 - 0.989i)T^{2} \)
23 \( 1 + (0.654 + 0.755i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (1.61 - 1.03i)T + (0.415 - 0.909i)T^{2} \)
37 \( 1 + 1.77T + T^{2} \)
41 \( 1 + (-0.723 - 0.690i)T^{2} \)
43 \( 1 + (0.239 + 1.66i)T + (-0.959 + 0.281i)T^{2} \)
47 \( 1 + (-0.981 + 0.189i)T^{2} \)
53 \( 1 + (-0.723 + 0.690i)T^{2} \)
59 \( 1 + (-0.580 - 0.814i)T^{2} \)
61 \( 1 + (1.11 - 0.326i)T + (0.841 - 0.540i)T^{2} \)
71 \( 1 + (-0.235 - 0.971i)T^{2} \)
73 \( 1 + (-0.341 + 1.40i)T + (-0.888 - 0.458i)T^{2} \)
79 \( 1 + (0.0845 - 1.77i)T + (-0.995 - 0.0950i)T^{2} \)
83 \( 1 + (0.888 - 0.458i)T^{2} \)
89 \( 1 + (-0.981 - 0.189i)T^{2} \)
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

−10.32896503404001155068685369420, −9.275906316155773214996220363678, −8.680337711011620853466274450213, −7.44258271761515744051174892450, −6.87116158519219828843686372420, −6.24291570162521176630757091797, −5.05525020680914467766101898660, −4.07125179305164709811475156745, −3.55939146137714555882209447855, −1.94829347355093365818959342475, 0.24221479891742602881477379122, 1.80323431074420474555964908769, 3.22502828996356562626763032703, 4.55284434427907326103022651121, 5.44481283995106374969281845792, 5.89488855073711603468924543697, 6.71073172791248505704151069200, 7.50251409117246986582634200347, 8.822752353128202432947420168603, 9.531107498801024700362064549656

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