Properties

Label 2-546-273.257-c1-0-34
Degree $2$
Conductor $546$
Sign $-0.711 - 0.702i$
Analytic cond. $4.35983$
Root an. cond. $2.08802$
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  − 2-s + (−0.733 − 1.56i)3-s + 4-s + (−3.72 − 2.14i)5-s + (0.733 + 1.56i)6-s + (1.54 − 2.14i)7-s − 8-s + (−1.92 + 2.30i)9-s + (3.72 + 2.14i)10-s + (2.26 − 3.91i)11-s + (−0.733 − 1.56i)12-s + (−2.01 − 2.98i)13-s + (−1.54 + 2.14i)14-s + (−0.639 + 7.41i)15-s + 16-s − 0.192·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.423 − 0.905i)3-s + 0.5·4-s + (−1.66 − 0.960i)5-s + (0.299 + 0.640i)6-s + (0.584 − 0.811i)7-s − 0.353·8-s + (−0.641 + 0.767i)9-s + (1.17 + 0.679i)10-s + (0.681 − 1.18i)11-s + (−0.211 − 0.452i)12-s + (−0.559 − 0.828i)13-s + (−0.413 + 0.573i)14-s + (−0.165 + 1.91i)15-s + 0.250·16-s − 0.0467·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.711 - 0.702i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.711 - 0.702i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.138510 + 0.337235i\)
\(L(\frac12)\) \(\approx\) \(0.138510 + 0.337235i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (0.733 + 1.56i)T \)
7 \( 1 + (-1.54 + 2.14i)T \)
13 \( 1 + (2.01 + 2.98i)T \)
good5 \( 1 + (3.72 + 2.14i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.26 + 3.91i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 0.192T + 17T^{2} \)
19 \( 1 + (0.845 + 1.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 1.40iT - 23T^{2} \)
29 \( 1 + (8.66 - 5.00i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.16 - 2.01i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 5.70iT - 37T^{2} \)
41 \( 1 + (-1.24 + 0.717i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.73 + 2.99i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-8.70 - 5.02i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.59 + 3.22i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 3.66iT - 59T^{2} \)
61 \( 1 + (7.02 - 4.05i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.38 + 2.52i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.59 - 9.68i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.90 + 10.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.93 - 3.34i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.15iT - 83T^{2} \)
89 \( 1 + 2.16iT - 89T^{2} \)
97 \( 1 + (-6.86 + 11.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.61393527437245664015525440203, −8.905891022654202955655063140405, −8.390014194402146823129180606897, −7.57353765692708416480331820728, −7.11594923731185868179553164200, −5.67388088301390283069467439502, −4.56038963370797453364763055199, −3.31966099596456524685156578779, −1.23258443781287742847415829526, −0.31800045393567901047776176652, 2.36544755021826504082226521024, 3.84375786435707139023731019869, 4.49865669504379593824409674505, 5.97285308290613902352332915704, 7.11497340581355648032539976813, 7.71691547525233153176355449358, 8.868502807697799907595951472918, 9.556657591721112402229116173623, 10.57652234224804952593721723166, 11.33445732991026621209807593237

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