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Properties

Label 2-546-273.173-c1-0-24
Degree $2$
Conductor $546$
Sign $0.696 - 0.717i$
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 + (1.25 − 1.19i)3-s + (−0.499 + 0.866i)4-s + (1.80 + 1.04i)5-s + (1.66 + 0.492i)6-s + (1.78 + 1.95i)7-s − 0.999·8-s + (0.160 − 2.99i)9-s + 2.08i·10-s − 2.15·11-s + (0.403 + 1.68i)12-s + (−0.217 + 3.59i)13-s + (−0.800 + 2.52i)14-s + (3.50 − 0.839i)15-s + (−0.5 − 0.866i)16-s + (−0.278 + 0.482i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.725 − 0.687i)3-s + (−0.249 + 0.433i)4-s + (0.806 + 0.465i)5-s + (0.677 + 0.201i)6-s + (0.674 + 0.738i)7-s − 0.353·8-s + (0.0534 − 0.998i)9-s + 0.658i·10-s − 0.650·11-s + (0.116 + 0.486i)12-s + (−0.0602 + 0.998i)13-s + (−0.213 + 0.673i)14-s + (0.905 − 0.216i)15-s + (−0.125 − 0.216i)16-s + (−0.0675 + 0.117i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.696 - 0.717i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.696 - 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.696 - 0.717i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.696 - 0.717i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.26658 + 0.959100i\)
\(L(\frac12)\) \(\approx\) \(2.26658 + 0.959100i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.25 + 1.19i)T \)
7 \( 1 + (-1.78 - 1.95i)T \)
13 \( 1 + (0.217 - 3.59i)T \)
good5 \( 1 + (-1.80 - 1.04i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + 2.15T + 11T^{2} \)
17 \( 1 + (0.278 - 0.482i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - 3.89T + 19T^{2} \)
23 \( 1 + (-1.79 + 1.03i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (6.10 + 3.52i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.21 + 5.57i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-7.20 + 4.15i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.532 - 0.307i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.33 - 7.50i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.507 + 0.292i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.68 - 3.85i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.35 - 0.782i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 8.20iT - 61T^{2} \)
67 \( 1 - 14.2iT - 67T^{2} \)
71 \( 1 + (6.52 + 11.3i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.198 + 0.344i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.73 + 9.93i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 12.4iT - 83T^{2} \)
89 \( 1 + (7.54 - 4.35i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.64 + 2.85i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.10473388725836316549952148296, −9.581100931787266490845492600717, −9.157356083186412702204451377264, −7.979210166888117995836626448875, −7.42888096503940798467444969774, −6.27914389909858897922485745216, −5.65514645792779935461370055980, −4.31991370956071161341288410694, −2.81755244189659220037511186437, −1.94249192754755690922584446783, 1.46615671624392944358250563099, 2.79703532267991054750864645333, 3.84630457900342020659162276848, 5.12172159053964069459482685123, 5.41400644798760375691391535277, 7.31790798984275560275235987719, 8.165497626737187792851217972183, 9.201645741688171554586238575551, 9.871618362228856643503506567977, 10.65413622385413756805847149301

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