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Properties

Label 2-546-91.80-c1-0-8
Degree $2$
Conductor $546$
Sign $0.880 + 0.473i$
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  + (−0.258 − 0.965i)2-s − i·3-s + (−0.866 + 0.499i)4-s + (−0.263 + 0.981i)5-s + (−0.965 + 0.258i)6-s + (2.62 + 0.355i)7-s + (0.707 + 0.707i)8-s − 9-s + 1.01·10-s + (2.99 + 2.99i)11-s + (0.499 + 0.866i)12-s + (−0.709 + 3.53i)13-s + (−0.335 − 2.62i)14-s + (0.981 + 0.263i)15-s + (0.500 − 0.866i)16-s + (−0.447 − 0.775i)17-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s − 0.577i·3-s + (−0.433 + 0.249i)4-s + (−0.117 + 0.439i)5-s + (−0.394 + 0.105i)6-s + (0.990 + 0.134i)7-s + (0.249 + 0.249i)8-s − 0.333·9-s + 0.321·10-s + (0.903 + 0.903i)11-s + (0.144 + 0.249i)12-s + (−0.196 + 0.980i)13-s + (−0.0896 − 0.701i)14-s + (0.253 + 0.0679i)15-s + (0.125 − 0.216i)16-s + (−0.108 − 0.188i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.880 + 0.473i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (535, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.880 + 0.473i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35931 - 0.342348i\)
\(L(\frac12)\) \(\approx\) \(1.35931 - 0.342348i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
3 \( 1 + iT \)
7 \( 1 + (-2.62 - 0.355i)T \)
13 \( 1 + (0.709 - 3.53i)T \)
good5 \( 1 + (0.263 - 0.981i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-2.99 - 2.99i)T + 11iT^{2} \)
17 \( 1 + (0.447 + 0.775i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.03 - 2.03i)T + 19iT^{2} \)
23 \( 1 + (-0.963 - 0.556i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.64 + 8.04i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.90 + 1.04i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-9.34 + 2.50i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (0.375 - 1.39i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (4.62 + 2.67i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.06 + 0.821i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.89 - 3.28i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (7.47 + 2.00i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + 2.35iT - 61T^{2} \)
67 \( 1 + (-3.05 + 3.05i)T - 67iT^{2} \)
71 \( 1 + (-0.229 - 0.858i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.120 - 0.449i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-3.14 - 5.45i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.53 + 7.53i)T + 83iT^{2} \)
89 \( 1 + (-3.39 - 12.6i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (8.06 - 2.16i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.02786618680419100036175138891, −9.771052612859155956625742975390, −9.146819175101047460527621160570, −8.002148578797377261188919091113, −7.29999117842495401531945675417, −6.29634441793778394918930292897, −4.87923561157187942832173756504, −3.94895603841461477105865870953, −2.41809800179884154366472960531, −1.44781602835406841031099334438, 1.05750429347591941440459392779, 3.22960671510921839174185274071, 4.51427117737039783502839567559, 5.20489497867013343755948320661, 6.21309078140682307788726921150, 7.40841386303974402743441242368, 8.386177546260750080888855601945, 8.835214671295379100252484342104, 9.862216223641351062537499801897, 10.88028350322662675342247199246

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