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Properties

Label 2-207-23.22-c6-0-18
Degree $2$
Conductor $207$
Sign $-0.990 - 0.139i$
Analytic cond. $47.6211$
Root an. cond. $6.90081$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.368·2-s − 63.8·4-s + 113. i·5-s + 569. i·7-s − 47.1·8-s + 41.9i·10-s + 814. i·11-s + 3.51e3·13-s + 210. i·14-s + 4.06e3·16-s + 7.48e3i·17-s + 4.28e3i·19-s − 7.27e3i·20-s + 300. i·22-s + (−1.20e4 − 1.69e3i)23-s + ⋯
L(s)  = 1  + 0.0460·2-s − 0.997·4-s + 0.911i·5-s + 1.66i·7-s − 0.0920·8-s + 0.0419i·10-s + 0.611i·11-s + 1.59·13-s + 0.0765i·14-s + 0.993·16-s + 1.52i·17-s + 0.624i·19-s − 0.909i·20-s + 0.0281i·22-s + (−0.990 − 0.139i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.139i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.990 - 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(207\)    =    \(3^{2} \cdot 23\)
Sign: $-0.990 - 0.139i$
Analytic conductor: \(47.6211\)
Root analytic conductor: \(6.90081\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{207} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 207,\ (\ :3),\ -0.990 - 0.139i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.467619879\)
\(L(\frac12)\) \(\approx\) \(1.467619879\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 + (1.20e4 + 1.69e3i)T \)
good2 \( 1 - 0.368T + 64T^{2} \)
5 \( 1 - 113. iT - 1.56e4T^{2} \)
7 \( 1 - 569. iT - 1.17e5T^{2} \)
11 \( 1 - 814. iT - 1.77e6T^{2} \)
13 \( 1 - 3.51e3T + 4.82e6T^{2} \)
17 \( 1 - 7.48e3iT - 2.41e7T^{2} \)
19 \( 1 - 4.28e3iT - 4.70e7T^{2} \)
29 \( 1 - 3.31e3T + 5.94e8T^{2} \)
31 \( 1 - 3.44e4T + 8.87e8T^{2} \)
37 \( 1 + 7.16e3iT - 2.56e9T^{2} \)
41 \( 1 + 7.79e4T + 4.75e9T^{2} \)
43 \( 1 - 1.13e5iT - 6.32e9T^{2} \)
47 \( 1 - 6.28e4T + 1.07e10T^{2} \)
53 \( 1 - 1.35e5iT - 2.21e10T^{2} \)
59 \( 1 - 1.01e5T + 4.21e10T^{2} \)
61 \( 1 + 3.82e4iT - 5.15e10T^{2} \)
67 \( 1 - 2.29e5iT - 9.04e10T^{2} \)
71 \( 1 - 6.45e4T + 1.28e11T^{2} \)
73 \( 1 + 5.36e4T + 1.51e11T^{2} \)
79 \( 1 + 4.58e5iT - 2.43e11T^{2} \)
83 \( 1 + 1.13e6iT - 3.26e11T^{2} \)
89 \( 1 - 8.29e5iT - 4.96e11T^{2} \)
97 \( 1 + 5.25e5iT - 8.32e11T^{2} \)
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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

−11.93265625544102892038872986705, −10.69323186852902928618119250994, −9.822946439262731334547052867947, −8.655181868262395318712753767742, −8.167415531593427972102544048949, −6.31473354266244716105734211569, −5.74992967248869042471035775806, −4.24199211797876367151239811892, −3.09783081835557386891873671701, −1.64725267387005330262268086622, 0.52313875128095708935262767201, 1.01045555304530276750156596996, 3.50497332103875045512691241757, 4.35107106484566501707788412118, 5.31781821771508079817922565760, 6.75757092434157868435312843202, 8.080576283149190969357110584048, 8.778338164842186100806419962888, 9.806305522541549609912989743948, 10.75668944364427725662105572877

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