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Properties

Label 2-546-273.257-c1-0-16
Degree $2$
Conductor $546$
Sign $0.904 - 0.425i$
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  + 2-s + (−1.72 + 0.148i)3-s + 4-s + (3.72 + 2.14i)5-s + (−1.72 + 0.148i)6-s + (1.54 − 2.14i)7-s + 8-s + (2.95 − 0.514i)9-s + (3.72 + 2.14i)10-s + (−2.26 + 3.91i)11-s + (−1.72 + 0.148i)12-s + (−2.01 − 2.98i)13-s + (1.54 − 2.14i)14-s + (−6.74 − 3.15i)15-s + 16-s + 0.192·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.996 + 0.0859i)3-s + 0.5·4-s + (1.66 + 0.960i)5-s + (−0.704 + 0.0608i)6-s + (0.584 − 0.811i)7-s + 0.353·8-s + (0.985 − 0.171i)9-s + (1.17 + 0.679i)10-s + (−0.681 + 1.18i)11-s + (−0.498 + 0.0429i)12-s + (−0.559 − 0.828i)13-s + (0.413 − 0.573i)14-s + (−1.74 − 0.814i)15-s + 0.250·16-s + 0.0467·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.425i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.904 - 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.904 - 0.425i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.904 - 0.425i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.14124 + 0.478794i\)
\(L(\frac12)\) \(\approx\) \(2.14124 + 0.478794i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (1.72 - 0.148i)T \)
7 \( 1 + (-1.54 + 2.14i)T \)
13 \( 1 + (2.01 + 2.98i)T \)
good5 \( 1 + (-3.72 - 2.14i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.26 - 3.91i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 - 0.192T + 17T^{2} \)
19 \( 1 + (0.845 + 1.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 1.40iT - 23T^{2} \)
29 \( 1 + (-8.66 + 5.00i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.16 - 2.01i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 5.70iT - 37T^{2} \)
41 \( 1 + (1.24 - 0.717i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.73 + 2.99i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (8.70 + 5.02i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.59 - 3.22i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 3.66iT - 59T^{2} \)
61 \( 1 + (7.02 - 4.05i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.38 + 2.52i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.59 + 9.68i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.90 + 10.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.93 - 3.34i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 4.15iT - 83T^{2} \)
89 \( 1 - 2.16iT - 89T^{2} \)
97 \( 1 + (-6.86 + 11.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.57616892272578927357079418665, −10.34136443863102460629277130934, −9.711967114456073226001563891291, −7.72806699875283398339622903845, −6.91176313323505750646215160020, −6.23336080453456215575191724773, −5.18935745576100277404346850974, −4.64384615483197649877147565904, −2.87653869071251196253274335163, −1.68263360241565158633845487642, 1.41677660259803577539016968716, 2.52855737801881626831200247054, 4.65583954232815306155404474748, 5.18052075213610586304379187902, 5.93357152354237734986102042891, 6.52749453416145830959535596918, 8.121331673511511647517337444996, 9.065439476622441901772646275091, 10.02002469334469876002468891427, 10.85504419586789181180527263088

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