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Properties

Label 2-546-91.19-c1-0-14
Degree $2$
Conductor $546$
Sign $-0.986 + 0.160i$
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  + (−0.965 − 0.258i)2-s − i·3-s + (0.866 + 0.499i)4-s + (−1.37 + 0.367i)5-s + (−0.258 + 0.965i)6-s + (−0.220 − 2.63i)7-s + (−0.707 − 0.707i)8-s − 9-s + 1.42·10-s + (2.72 + 2.72i)11-s + (0.499 − 0.866i)12-s + (−1.74 − 3.15i)13-s + (−0.469 + 2.60i)14-s + (0.367 + 1.37i)15-s + (0.500 + 0.866i)16-s + (1.18 − 2.04i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s − 0.577i·3-s + (0.433 + 0.249i)4-s + (−0.614 + 0.164i)5-s + (−0.105 + 0.394i)6-s + (−0.0834 − 0.996i)7-s + (−0.249 − 0.249i)8-s − 0.333·9-s + 0.449·10-s + (0.822 + 0.822i)11-s + (0.144 − 0.249i)12-s + (−0.484 − 0.875i)13-s + (−0.125 + 0.695i)14-s + (0.0949 + 0.354i)15-s + (0.125 + 0.216i)16-s + (0.286 − 0.495i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.160i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 + 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.986 + 0.160i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.986 + 0.160i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0368970 - 0.455391i\)
\(L(\frac12)\) \(\approx\) \(0.0368970 - 0.455391i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
3 \( 1 + iT \)
7 \( 1 + (0.220 + 2.63i)T \)
13 \( 1 + (1.74 + 3.15i)T \)
good5 \( 1 + (1.37 - 0.367i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-2.72 - 2.72i)T + 11iT^{2} \)
17 \( 1 + (-1.18 + 2.04i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.27 + 4.27i)T + 19iT^{2} \)
23 \( 1 + (7.02 - 4.05i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.220 - 0.382i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.855 + 3.19i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-1.96 + 7.32i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.33 - 0.357i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (7.53 - 4.35i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.64 + 6.14i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (4.08 + 7.07i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.17 - 8.11i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 - 10.6iT - 61T^{2} \)
67 \( 1 + (0.0278 - 0.0278i)T - 67iT^{2} \)
71 \( 1 + (12.2 + 3.28i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-7.72 - 2.06i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-6.14 + 10.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (9.37 + 9.37i)T + 83iT^{2} \)
89 \( 1 + (-1.99 - 0.533i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-3.62 + 13.5i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.25126405568610771697368176945, −9.662898076219129304374564230786, −8.486988027676931060256577139334, −7.45690932801049744059721824234, −7.25616112188215575961098178740, −6.07730020724581588781744891235, −4.47579518497133608431111691994, −3.41587817952169542839508221552, −1.92480308781920448947580348613, −0.31625502701333721770213918979, 1.97796174959310840489048551344, 3.52677952425482159510785794945, 4.56637394961635371378396501852, 5.97191599419226954635941266958, 6.52062736821075247429013324379, 8.229450542702901519456715879629, 8.362587450474798430012099074912, 9.439133663732912682513418694186, 10.14124967669864034049914901928, 11.19486036240220378840411488401

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