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Properties

Label 2-546-273.44-c1-0-18
Degree $2$
Conductor $546$
Sign $0.419 + 0.907i$
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 + (−1.45 + 0.945i)3-s + (0.866 − 0.499i)4-s + (−2.31 + 0.619i)5-s + (−1.15 + 1.28i)6-s + (−1.72 − 2.00i)7-s + (0.707 − 0.707i)8-s + (1.21 − 2.74i)9-s + (−2.07 + 1.19i)10-s + (4.27 + 1.14i)11-s + (−0.783 + 1.54i)12-s + (2.93 − 2.08i)13-s + (−2.18 − 1.48i)14-s + (2.77 − 3.08i)15-s + (0.500 − 0.866i)16-s + (−0.783 − 1.35i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (−0.837 + 0.546i)3-s + (0.433 − 0.249i)4-s + (−1.03 + 0.277i)5-s + (−0.472 + 0.526i)6-s + (−0.653 − 0.756i)7-s + (0.249 − 0.249i)8-s + (0.403 − 0.915i)9-s + (−0.655 + 0.378i)10-s + (1.28 + 0.345i)11-s + (−0.226 + 0.445i)12-s + (0.814 − 0.579i)13-s + (−0.584 − 0.397i)14-s + (0.715 − 0.797i)15-s + (0.125 − 0.216i)16-s + (−0.189 − 0.328i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.419 + 0.907i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.419 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.419 + 0.907i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.419 + 0.907i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03333 - 0.660718i\)
\(L(\frac12)\) \(\approx\) \(1.03333 - 0.660718i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.45 - 0.945i)T \)
7 \( 1 + (1.72 + 2.00i)T \)
13 \( 1 + (-2.93 + 2.08i)T \)
good5 \( 1 + (2.31 - 0.619i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-4.27 - 1.14i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.783 + 1.35i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.62 + 6.07i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.38 + 2.40i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 10.1iT - 29T^{2} \)
31 \( 1 + (-3.55 - 0.952i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-3.11 + 0.834i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (3.20 + 3.20i)T + 41iT^{2} \)
43 \( 1 - 4.41iT - 43T^{2} \)
47 \( 1 + (-2.70 - 10.0i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.99 - 1.15i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (13.4 + 3.59i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-6.28 + 10.8i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.94 - 0.520i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (2.32 + 2.32i)T + 71iT^{2} \)
73 \( 1 + (0.292 - 1.09i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (1.69 - 2.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.01 + 3.01i)T + 83iT^{2} \)
89 \( 1 + (1.07 + 4.01i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-1.97 + 1.97i)T - 97iT^{2} \)
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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

−11.06540860610339275170571179030, −9.990621940996557947987070186228, −9.140464754658409399632347668546, −7.67483507081267450732580100002, −6.65390624106891454932665123844, −6.20392851693893920503232207234, −4.59380069347033626111426086424, −4.10104233636530657796372820993, −3.16302691943311956948496497300, −0.68808966475697287150310828621, 1.54709378768431687004709300759, 3.47378371451954952226508502353, 4.27690797124835809080676253248, 5.58689943868539855286430182061, 6.33320062064337391599280235497, 7.01722894296163383111763302350, 8.215336698120973648818953892387, 8.943255766079620074208467223926, 10.38103884428304884954882194529, 11.44514541797423508068496720441

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