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Properties

Label 2-546-91.34-c1-0-9
Degree $2$
Conductor $546$
Sign $0.985 + 0.169i$
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 + i·3-s + 1.00i·4-s + (0.461 − 0.461i)5-s + (0.707 − 0.707i)6-s + (−2.62 + 0.292i)7-s + (0.707 − 0.707i)8-s − 9-s − 0.652·10-s + (1.60 − 1.60i)11-s − 1.00·12-s + (2.51 − 2.58i)13-s + (2.06 + 1.65i)14-s + (0.461 + 0.461i)15-s − 1.00·16-s + 6.40·17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.577i·3-s + 0.500i·4-s + (0.206 − 0.206i)5-s + (0.288 − 0.288i)6-s + (−0.993 + 0.110i)7-s + (0.250 − 0.250i)8-s − 0.333·9-s − 0.206·10-s + (0.483 − 0.483i)11-s − 0.288·12-s + (0.697 − 0.716i)13-s + (0.552 + 0.441i)14-s + (0.119 + 0.119i)15-s − 0.250·16-s + 1.55·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.985 + 0.169i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.985 + 0.169i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12188 - 0.0956139i\)
\(L(\frac12)\) \(\approx\) \(1.12188 - 0.0956139i\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
7 \( 1 + (2.62 - 0.292i)T \)
13 \( 1 + (-2.51 + 2.58i)T \)
good5 \( 1 + (-0.461 + 0.461i)T - 5iT^{2} \)
11 \( 1 + (-1.60 + 1.60i)T - 11iT^{2} \)
17 \( 1 - 6.40T + 17T^{2} \)
19 \( 1 + (-2.52 + 2.52i)T - 19iT^{2} \)
23 \( 1 - 8.51iT - 23T^{2} \)
29 \( 1 - 7.91T + 29T^{2} \)
31 \( 1 + (-0.922 + 0.922i)T - 31iT^{2} \)
37 \( 1 + (0.953 - 0.953i)T - 37iT^{2} \)
41 \( 1 + (3.89 - 3.89i)T - 41iT^{2} \)
43 \( 1 + 4.17iT - 43T^{2} \)
47 \( 1 + (-1.41 - 1.41i)T + 47iT^{2} \)
53 \( 1 - 9.10T + 53T^{2} \)
59 \( 1 + (-1.17 - 1.17i)T + 59iT^{2} \)
61 \( 1 + 14.4iT - 61T^{2} \)
67 \( 1 + (-3.47 - 3.47i)T + 67iT^{2} \)
71 \( 1 + (8.29 + 8.29i)T + 71iT^{2} \)
73 \( 1 + (5.89 + 5.89i)T + 73iT^{2} \)
79 \( 1 + 9.18T + 79T^{2} \)
83 \( 1 + (-9.98 + 9.98i)T - 83iT^{2} \)
89 \( 1 + (-8.54 - 8.54i)T + 89iT^{2} \)
97 \( 1 + (-0.272 + 0.272i)T - 97iT^{2} \)
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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

−10.57856673944800044734376743992, −9.827826596629391651496533138508, −9.281556737115416964723941274050, −8.385620375132066346862115447628, −7.33974269233754916607018296144, −6.06995512096730887084979933883, −5.23637185987441978932645505119, −3.55946022736346593788122226841, −3.13983751760370926054878140267, −1.08502958334600674661382483690, 1.09664939771924606348323149099, 2.72202364009245272597172537559, 4.12882491463326027231597007463, 5.67287607680746717034166388331, 6.51522810687223321267744157057, 7.01522979849775686638992677941, 8.177837412201652608288994052383, 8.936048614353554142924179374325, 10.03525567309087473690270421928, 10.39474307313798366192302997084

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