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Properties

Label 2-1407-1407.1061-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: $\chi_{1407} (1061, \cdot )$
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

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

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