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Properties

Label 2-2667-2667.1277-c0-0-0
Degree $2$
Conductor $2667$
Sign $0.947 - 0.320i$
Analytic cond. $1.33100$
Root an. cond. $1.15369$
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  + (−0.969 − 0.246i)3-s + (0.955 + 0.294i)4-s + (−0.766 − 0.642i)7-s + (0.878 + 0.478i)9-s + (−0.853 − 0.521i)12-s + (0.297 + 0.393i)13-s + (0.826 + 0.563i)16-s + 1.49i·19-s + (0.583 + 0.811i)21-s + (0.0747 + 0.997i)25-s + (−0.733 − 0.680i)27-s + (−0.542 − 0.840i)28-s + (−0.177 − 0.0908i)31-s + (0.698 + 0.715i)36-s + (1.22 − 1.02i)37-s + ⋯
L(s)  = 1  + (−0.969 − 0.246i)3-s + (0.955 + 0.294i)4-s + (−0.766 − 0.642i)7-s + (0.878 + 0.478i)9-s + (−0.853 − 0.521i)12-s + (0.297 + 0.393i)13-s + (0.826 + 0.563i)16-s + 1.49i·19-s + (0.583 + 0.811i)21-s + (0.0747 + 0.997i)25-s + (−0.733 − 0.680i)27-s + (−0.542 − 0.840i)28-s + (−0.177 − 0.0908i)31-s + (0.698 + 0.715i)36-s + (1.22 − 1.02i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2667\)    =    \(3 \cdot 7 \cdot 127\)
Sign: $0.947 - 0.320i$
Analytic conductor: \(1.33100\)
Root analytic conductor: \(1.15369\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2667} (1277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2667,\ (\ :0),\ 0.947 - 0.320i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.056985108\)
\(L(\frac12)\) \(\approx\) \(1.056985108\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.969 + 0.246i)T \)
7 \( 1 + (0.766 + 0.642i)T \)
127 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (-0.955 - 0.294i)T^{2} \)
5 \( 1 + (-0.0747 - 0.997i)T^{2} \)
11 \( 1 + (0.124 + 0.992i)T^{2} \)
13 \( 1 + (-0.297 - 0.393i)T + (-0.270 + 0.962i)T^{2} \)
17 \( 1 + (-0.583 + 0.811i)T^{2} \)
19 \( 1 - 1.49iT - T^{2} \)
23 \( 1 + (-0.124 + 0.992i)T^{2} \)
29 \( 1 + (-0.661 - 0.749i)T^{2} \)
31 \( 1 + (0.177 + 0.0908i)T + (0.583 + 0.811i)T^{2} \)
37 \( 1 + (-1.22 + 1.02i)T + (0.173 - 0.984i)T^{2} \)
41 \( 1 + (0.995 + 0.0995i)T^{2} \)
43 \( 1 + (-1.51 + 1.08i)T + (0.318 - 0.947i)T^{2} \)
47 \( 1 + (0.826 + 0.563i)T^{2} \)
53 \( 1 + (0.456 - 0.889i)T^{2} \)
59 \( 1 + (-0.173 + 0.984i)T^{2} \)
61 \( 1 + (-0.0994 - 0.00744i)T + (0.988 + 0.149i)T^{2} \)
67 \( 1 + (0.658 - 0.185i)T + (0.853 - 0.521i)T^{2} \)
71 \( 1 + (0.797 + 0.603i)T^{2} \)
73 \( 1 + (-1.86 - 0.730i)T + (0.733 + 0.680i)T^{2} \)
79 \( 1 + (-0.110 + 0.329i)T + (-0.797 - 0.603i)T^{2} \)
83 \( 1 + (0.998 - 0.0498i)T^{2} \)
89 \( 1 + (0.222 + 0.974i)T^{2} \)
97 \( 1 + (0.148 + 0.0148i)T + (0.980 + 0.198i)T^{2} \)
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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

−9.223829190122126154704160773641, −7.949705305173747183237365639541, −7.40766818109072897236927076437, −6.77802606463735353027924436332, −6.05006205287772552049029764775, −5.54557264498684568611918490202, −4.14961277514585596617548161832, −3.57595062002141092099631368792, −2.26659830988954524159393691994, −1.18955049562838761134038357067, 0.898849933768470370463972887188, 2.38490577852099340818041687973, 3.16097273336963947885492794886, 4.41171374029173949101783022537, 5.28153494981006555316706281158, 6.09654436191666647205379251113, 6.46235218921479427213958358031, 7.20628682954509808780388896057, 8.157480762165715373570014408420, 9.284792298103599397390040917266

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