Properties

Label 2-546-91.33-c1-0-10
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: $\chi_{546} (397, \cdot )$
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

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

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