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Properties

Label 2-546-7.4-c1-0-11
Degree $2$
Conductor $546$
Sign $0.832 + 0.553i$
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  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.822 − 1.42i)5-s − 0.999·6-s + (−1.32 − 2.29i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (0.822 − 1.42i)10-s + (2.32 − 4.02i)11-s + (−0.499 − 0.866i)12-s + 13-s + (1.32 − 2.29i)14-s + 1.64·15-s + (−0.5 − 0.866i)16-s + (0.677 − 1.17i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.368 − 0.637i)5-s − 0.408·6-s + (−0.499 − 0.866i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.260 − 0.450i)10-s + (0.700 − 1.21i)11-s + (−0.144 − 0.249i)12-s + 0.277·13-s + (0.353 − 0.612i)14-s + 0.424·15-s + (−0.125 − 0.216i)16-s + (0.164 − 0.284i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.832 + 0.553i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.832 + 0.553i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10953 - 0.335318i\)
\(L(\frac12)\) \(\approx\) \(1.10953 - 0.335318i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (1.32 + 2.29i)T \)
13 \( 1 - T \)
good5 \( 1 + (0.822 + 1.42i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.32 + 4.02i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.677 + 1.17i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.5 + 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.82 - 6.62i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.29T + 29T^{2} \)
31 \( 1 + (-3.64 + 6.31i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.177 + 0.306i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.64T + 41T^{2} \)
43 \( 1 - 5.29T + 43T^{2} \)
47 \( 1 + (1.5 + 2.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.96 - 6.87i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.96 + 3.40i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.79 + 11.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.70T + 71T^{2} \)
73 \( 1 + (-4.17 + 7.23i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5 - 8.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.70T + 83T^{2} \)
89 \( 1 + (0.531 + 0.920i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 14.9T + 97T^{2} \)
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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

−10.94680061748840813455932778561, −9.608541293734710803433654361863, −8.970708621945843055864988466078, −8.022294605920983594781481022023, −6.95561172158624894000386698643, −6.10517356110344905167770718797, −5.08019389082495732808441322108, −4.07485586424331994408824021940, −3.33795461230799945310578218853, −0.64390904722481693073505148319, 1.71884071920945918057169206292, 2.90656256195210699183352819782, 4.05347376052393206912354134315, 5.28435990065643125101777616202, 6.40738383050642747397524253965, 6.97537806748975444541489141656, 8.298262675993771684082722686936, 9.245221981300529958952985408587, 10.21813159385914255645857762354, 10.98477998754956840732982426021

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