Properties

Label 2-207-207.11-c1-0-12
Degree $2$
Conductor $207$
Sign $-0.632 + 0.774i$
Analytic cond. $1.65290$
Root an. cond. $1.28565$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.53 + 0.0731i)2-s + (−0.597 − 1.62i)3-s + (0.359 − 0.0343i)4-s + (1.46 + 1.39i)5-s + (1.03 + 2.45i)6-s + (−0.635 − 3.29i)7-s + (2.49 − 0.358i)8-s + (−2.28 + 1.94i)9-s + (−2.34 − 2.03i)10-s + (2.41 − 1.24i)11-s + (−0.270 − 0.564i)12-s + (−5.49 − 1.05i)13-s + (1.21 + 5.01i)14-s + (1.39 − 3.20i)15-s + (−4.50 + 0.869i)16-s + (−2.19 − 4.80i)17-s + ⋯
L(s)  = 1  + (−1.08 + 0.0517i)2-s + (−0.345 − 0.938i)3-s + (0.179 − 0.0171i)4-s + (0.653 + 0.622i)5-s + (0.423 + 1.00i)6-s + (−0.240 − 1.24i)7-s + (0.881 − 0.126i)8-s + (−0.761 + 0.647i)9-s + (−0.740 − 0.642i)10-s + (0.728 − 0.375i)11-s + (−0.0782 − 0.162i)12-s + (−1.52 − 0.293i)13-s + (0.325 + 1.34i)14-s + (0.359 − 0.827i)15-s + (−1.12 + 0.217i)16-s + (−0.532 − 1.16i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.632 + 0.774i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.632 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(207\)    =    \(3^{2} \cdot 23\)
Sign: $-0.632 + 0.774i$
Analytic conductor: \(1.65290\)
Root analytic conductor: \(1.28565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{207} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 207,\ (\ :1/2),\ -0.632 + 0.774i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.194713 - 0.410102i\)
\(L(\frac12)\) \(\approx\) \(0.194713 - 0.410102i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.597 + 1.62i)T \)
23 \( 1 + (4.60 + 1.32i)T \)
good2 \( 1 + (1.53 - 0.0731i)T + (1.99 - 0.190i)T^{2} \)
5 \( 1 + (-1.46 - 1.39i)T + (0.237 + 4.99i)T^{2} \)
7 \( 1 + (0.635 + 3.29i)T + (-6.49 + 2.60i)T^{2} \)
11 \( 1 + (-2.41 + 1.24i)T + (6.38 - 8.96i)T^{2} \)
13 \( 1 + (5.49 + 1.05i)T + (12.0 + 4.83i)T^{2} \)
17 \( 1 + (2.19 + 4.80i)T + (-11.1 + 12.8i)T^{2} \)
19 \( 1 + (2.07 + 0.948i)T + (12.4 + 14.3i)T^{2} \)
29 \( 1 + (0.462 - 4.84i)T + (-28.4 - 5.48i)T^{2} \)
31 \( 1 + (-7.61 + 5.98i)T + (7.30 - 30.1i)T^{2} \)
37 \( 1 + (-2.60 + 8.88i)T + (-31.1 - 20.0i)T^{2} \)
41 \( 1 + (2.00 - 2.10i)T + (-1.95 - 40.9i)T^{2} \)
43 \( 1 + (3.99 - 5.07i)T + (-10.1 - 41.7i)T^{2} \)
47 \( 1 + (-8.27 - 4.78i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.39 - 2.76i)T + (-7.54 + 52.4i)T^{2} \)
59 \( 1 + (0.0737 - 0.382i)T + (-54.7 - 21.9i)T^{2} \)
61 \( 1 + (1.33 - 3.34i)T + (-44.1 - 42.0i)T^{2} \)
67 \( 1 + (-2.70 + 5.25i)T + (-38.8 - 54.5i)T^{2} \)
71 \( 1 + (2.11 + 3.29i)T + (-29.4 + 64.5i)T^{2} \)
73 \( 1 + (0.822 - 1.80i)T + (-47.8 - 55.1i)T^{2} \)
79 \( 1 + (0.701 - 0.242i)T + (62.0 - 48.8i)T^{2} \)
83 \( 1 + (-9.78 + 9.33i)T + (3.94 - 82.9i)T^{2} \)
89 \( 1 + (-0.937 + 6.51i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (-13.6 - 3.32i)T + (86.2 + 44.4i)T^{2} \)
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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

−11.89212476427808502397048636129, −10.78649926378041791808411660933, −10.09792112079520347031409000061, −9.176007355235269023738688121729, −7.79710282363838304247750757788, −7.14560834694767793130644688825, −6.31060617191909995745837524366, −4.56265277838118860502393051084, −2.38870050882716548589617381246, −0.56906937813241421929184082509, 2.08550846086362579606784265669, 4.33474149662586156630399271078, 5.34228063462996795228917447353, 6.52019489240929510238169924575, 8.344007031822767474411012079857, 8.994852067241696847007876556587, 9.778882487814021954489227567220, 10.23432232924347254255853871526, 11.74560514607180216312832299182, 12.35545478484054407820323723619

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