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Properties

Label 2-1206-67.29-c1-0-9
Degree $2$
Conductor $1206$
Sign $-0.224 - 0.974i$
Analytic cond. $9.62995$
Root an. cond. $3.10321$
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  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + 2.69·5-s + (0.555 + 0.962i)7-s + 0.999·8-s + (−1.34 + 2.33i)10-s + (0.738 + 1.27i)11-s + (−2.43 + 4.22i)13-s − 1.11·14-s + (−0.5 + 0.866i)16-s + (1.34 − 2.33i)17-s + (−3.78 + 6.56i)19-s + (−1.34 − 2.33i)20-s − 1.47·22-s + (0.349 − 0.605i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + 1.20·5-s + (0.210 + 0.363i)7-s + 0.353·8-s + (−0.426 + 0.739i)10-s + (0.222 + 0.385i)11-s + (−0.676 + 1.17i)13-s − 0.296·14-s + (−0.125 + 0.216i)16-s + (0.327 − 0.567i)17-s + (−0.869 + 1.50i)19-s + (−0.301 − 0.522i)20-s − 0.314·22-s + (0.0729 − 0.126i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1206 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.224 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1206 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.224 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1206\)    =    \(2 \cdot 3^{2} \cdot 67\)
Sign: $-0.224 - 0.974i$
Analytic conductor: \(9.62995\)
Root analytic conductor: \(3.10321\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1206} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1206,\ (\ :1/2),\ -0.224 - 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.535180254\)
\(L(\frac12)\) \(\approx\) \(1.535180254\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
67 \( 1 + (7.74 - 2.65i)T \)
good5 \( 1 - 2.69T + 5T^{2} \)
7 \( 1 + (-0.555 - 0.962i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.738 - 1.27i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.43 - 4.22i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.34 + 2.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.78 - 6.56i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.349 + 0.605i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.08 - 1.88i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.905 - 1.56i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.64 + 2.84i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.89 - 5.02i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 1.36T + 43T^{2} \)
47 \( 1 + (-2.49 - 4.31i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 2.51T + 53T^{2} \)
59 \( 1 - 7.32T + 59T^{2} \)
61 \( 1 + (-2.93 + 5.08i)T + (-30.5 - 52.8i)T^{2} \)
71 \( 1 + (0.0716 + 0.124i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.78 - 10.0i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.88 - 6.72i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.75 + 9.96i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 5.47T + 89T^{2} \)
97 \( 1 + (-5.71 + 9.90i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.831591246567385798265791493637, −9.205423781903823110188313678074, −8.452144003829944310074560462334, −7.43335239977921495773938906717, −6.60343548793729695696497115144, −5.90794755156825269411415242527, −5.09615803876628685817040679670, −4.12681346965650568310174402404, −2.41139830495574845249492365255, −1.56602964997759572200359646926, 0.75476676594503074347577235173, 2.11169325094716409280792735542, 2.94146997571660491929820541917, 4.23099826025276484987078427588, 5.26324298156142361796994363211, 6.08586122118517517601768543155, 7.10337621518899677885853060502, 8.046991529870444136826398889489, 8.885018079449889485715889678402, 9.606288580538929759644895474107

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