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Properties

Label 2-546-91.45-c1-0-12
Degree $2$
Conductor $546$
Sign $0.608 + 0.793i$
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  + (−0.707 + 0.707i)2-s + (0.866 − 0.5i)3-s − 1.00i·4-s + (0.756 − 2.82i)5-s + (−0.258 + 0.965i)6-s + (2.64 + 0.162i)7-s + (0.707 + 0.707i)8-s + (0.499 − 0.866i)9-s + (1.46 + 2.53i)10-s + (0.897 − 3.34i)11-s + (−0.500 − 0.866i)12-s + (−3.36 + 1.28i)13-s + (−1.98 + 1.75i)14-s + (−0.756 − 2.82i)15-s − 1.00·16-s + 1.24·17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.499 − 0.288i)3-s − 0.500i·4-s + (0.338 − 1.26i)5-s + (−0.105 + 0.394i)6-s + (0.998 + 0.0614i)7-s + (0.250 + 0.250i)8-s + (0.166 − 0.288i)9-s + (0.462 + 0.800i)10-s + (0.270 − 1.00i)11-s + (−0.144 − 0.250i)12-s + (−0.933 + 0.357i)13-s + (−0.529 + 0.468i)14-s + (−0.195 − 0.729i)15-s − 0.250·16-s + 0.302·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.608 + 0.793i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.608 + 0.793i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31830 - 0.650533i\)
\(L(\frac12)\) \(\approx\) \(1.31830 - 0.650533i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (-2.64 - 0.162i)T \)
13 \( 1 + (3.36 - 1.28i)T \)
good5 \( 1 + (-0.756 + 2.82i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.897 + 3.34i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 - 1.24T + 17T^{2} \)
19 \( 1 + (4.22 - 1.13i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + 1.33iT - 23T^{2} \)
29 \( 1 + (-0.896 + 1.55i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.65 + 0.443i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (2.73 + 2.73i)T + 37iT^{2} \)
41 \( 1 + (-2.20 + 0.589i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (4.56 - 2.63i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-11.1 - 2.98i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.96 - 5.14i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7.03 + 7.03i)T - 59iT^{2} \)
61 \( 1 + (-6.72 - 3.88i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.35 - 0.898i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (9.20 + 2.46i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-4.15 - 15.5i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (3.83 + 6.64i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.31 - 2.31i)T + 83iT^{2} \)
89 \( 1 + (-5.26 + 5.26i)T - 89iT^{2} \)
97 \( 1 + (-2.79 + 10.4i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.49029527868134692312639920415, −9.448724232567157147491651770028, −8.676457123056274141087045990419, −8.277292566263874670637368560266, −7.30792170043951710276877363052, −6.07718193372612692818669463797, −5.15136203045232933236329841680, −4.20699850668877437319541122373, −2.23798246209516189441014186637, −1.01758275444649620024364732423, 1.93563385009288609340258075403, 2.74199491991670078322126788937, 4.05149806978280689755444312646, 5.14033112735617323742252120215, 6.75707029960528405586475128988, 7.42646024818780441120501863489, 8.318231231470486751010766323022, 9.347266243446247974375433335412, 10.27456788666969591392239416109, 10.56477064159205719015449230313

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