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Properties

Label 2-207-207.160-c0-0-0
Degree $2$
Conductor $207$
Sign $-0.5 - 0.866i$
Analytic cond. $0.103306$
Root an. cond. $0.321413$
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  + (−0.766 + 1.32i)2-s + (0.766 + 0.642i)3-s + (−0.673 − 1.16i)4-s + (−1.43 + 0.524i)6-s + 0.532·8-s + (0.173 + 0.984i)9-s + (0.233 − 1.32i)12-s + (−0.766 − 1.32i)13-s + (0.266 − 0.460i)16-s + (−1.43 − 0.524i)18-s + (−0.5 − 0.866i)23-s + (0.407 + 0.342i)24-s + (−0.5 + 0.866i)25-s + 2.34·26-s + (−0.500 + 0.866i)27-s + ⋯
L(s)  = 1  + (−0.766 + 1.32i)2-s + (0.766 + 0.642i)3-s + (−0.673 − 1.16i)4-s + (−1.43 + 0.524i)6-s + 0.532·8-s + (0.173 + 0.984i)9-s + (0.233 − 1.32i)12-s + (−0.766 − 1.32i)13-s + (0.266 − 0.460i)16-s + (−1.43 − 0.524i)18-s + (−0.5 − 0.866i)23-s + (0.407 + 0.342i)24-s + (−0.5 + 0.866i)25-s + 2.34·26-s + (−0.500 + 0.866i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(207\)    =    \(3^{2} \cdot 23\)
Sign: $-0.5 - 0.866i$
Analytic conductor: \(0.103306\)
Root analytic conductor: \(0.321413\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{207} (160, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 207,\ (\ :0),\ -0.5 - 0.866i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5817844596\)
\(L(\frac12)\) \(\approx\) \(0.5817844596\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.766 - 0.642i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
29 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + 1.87T + T^{2} \)
73 \( 1 + 1.87T + T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−13.31449246953380975095121742173, −12.00829910333914853742164124314, −10.32137925947366348543295774805, −9.897206494457162580832771227658, −8.693095989870929723010969814932, −8.076759748930071063219585044455, −7.19170087334879226206411709511, −5.83323076343614519974055294283, −4.70521179965447883351608023496, −2.93369459737726509413576750423, 1.71818216672499630298919976284, 2.83838046183866168281928179967, 4.20910329994878808624538866646, 6.37725458535952693750632654202, 7.62153136794229150128735649942, 8.625958467544191874718240478068, 9.464200926413797679301120810734, 10.15804413881525896044634225712, 11.58952195766698265233548143128, 12.01169495722478337966997379485

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