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Properties

Label 2-546-39.2-c1-0-10
Degree $2$
Conductor $546$
Sign $0.985 + 0.172i$
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 + (−1.34 − 1.08i)3-s + (−0.866 − 0.499i)4-s + (0.429 + 0.429i)5-s + (1.39 − 1.02i)6-s + (−0.965 + 0.258i)7-s + (0.707 − 0.707i)8-s + (0.644 + 2.92i)9-s + (−0.526 + 0.303i)10-s + (1.44 + 0.388i)11-s + (0.626 + 1.61i)12-s + (−1.82 − 3.11i)13-s − i·14-s + (−0.113 − 1.04i)15-s + (0.500 + 0.866i)16-s + (2.50 − 4.33i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−0.779 − 0.626i)3-s + (−0.433 − 0.249i)4-s + (0.192 + 0.192i)5-s + (0.570 − 0.417i)6-s + (−0.365 + 0.0978i)7-s + (0.249 − 0.249i)8-s + (0.214 + 0.976i)9-s + (−0.166 + 0.0960i)10-s + (0.436 + 0.117i)11-s + (0.180 + 0.466i)12-s + (−0.505 − 0.862i)13-s − 0.267i·14-s + (−0.0293 − 0.270i)15-s + (0.125 + 0.216i)16-s + (0.606 − 1.05i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.172i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.985 + 0.172i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.985 + 0.172i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.939161 - 0.0813870i\)
\(L(\frac12)\) \(\approx\) \(0.939161 - 0.0813870i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.34 + 1.08i)T \)
7 \( 1 + (0.965 - 0.258i)T \)
13 \( 1 + (1.82 + 3.11i)T \)
good5 \( 1 + (-0.429 - 0.429i)T + 5iT^{2} \)
11 \( 1 + (-1.44 - 0.388i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-2.50 + 4.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.744 - 2.77i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-2.87 - 4.97i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.53 + 3.19i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.86 + 2.86i)T - 31iT^{2} \)
37 \( 1 + (-2.29 + 8.56i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.973 + 3.63i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-9.35 - 5.39i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-7.10 + 7.10i)T - 47iT^{2} \)
53 \( 1 - 7.88iT - 53T^{2} \)
59 \( 1 + (-3.35 - 12.5i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.879 - 1.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.7 + 2.87i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (5.54 - 1.48i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (10.8 + 10.8i)T + 73iT^{2} \)
79 \( 1 - 3.68T + 79T^{2} \)
83 \( 1 + (5.15 + 5.15i)T + 83iT^{2} \)
89 \( 1 + (-6.57 - 1.76i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (0.785 + 2.92i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.60640043679228490038873384091, −9.952444789602710355268368878157, −8.984895387952033887051119361020, −7.62450678186545025249671703348, −7.33306289807387594275381896188, −6.07755967755655501363251988860, −5.62284504540613614622840262667, −4.42045534372986115323919117237, −2.68056582485744999975943568044, −0.812687340131946737170521902249, 1.15999222581680790061280880051, 2.99362884486956079261139225789, 4.18572346557555800784368140016, 4.98015254758219127224128306202, 6.18943880524840956303834522407, 7.04335375925348065974508675225, 8.571816997590962262773917206467, 9.302420985244657778577855388823, 10.05896998277911436438284273783, 10.77592217093493060944955457126

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