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Properties

Label 2-546-39.5-c1-0-25
Degree $2$
Conductor $546$
Sign $0.387 + 0.921i$
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 + (1.41 − i)3-s + 1.00i·4-s + (−2.47 − 2.47i)5-s + (1.70 + 0.292i)6-s + (−0.707 − 0.707i)7-s + (−0.707 + 0.707i)8-s + (1.00 − 2.82i)9-s − 3.49i·10-s + (0.976 − 0.976i)11-s + (1.00 + 1.41i)12-s + (1.76 − 3.14i)13-s − 1.00i·14-s + (−5.96 − 1.02i)15-s − 1.00·16-s − 5.49·17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.816 − 0.577i)3-s + 0.500i·4-s + (−1.10 − 1.10i)5-s + (0.696 + 0.119i)6-s + (−0.267 − 0.267i)7-s + (−0.250 + 0.250i)8-s + (0.333 − 0.942i)9-s − 1.10i·10-s + (0.294 − 0.294i)11-s + (0.288 + 0.408i)12-s + (0.489 − 0.872i)13-s − 0.267i·14-s + (−1.54 − 0.264i)15-s − 0.250·16-s − 1.33·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.387 + 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.387 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.387 + 0.921i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.387 + 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56274 - 1.03780i\)
\(L(\frac12)\) \(\approx\) \(1.56274 - 1.03780i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.41 + i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + (-1.76 + 3.14i)T \)
good5 \( 1 + (2.47 + 2.47i)T + 5iT^{2} \)
11 \( 1 + (-0.976 + 0.976i)T - 11iT^{2} \)
17 \( 1 + 5.49T + 17T^{2} \)
19 \( 1 + (-0.470 + 0.470i)T - 19iT^{2} \)
23 \( 1 - 8.97T + 23T^{2} \)
29 \( 1 + 2.03iT - 29T^{2} \)
31 \( 1 + (-0.585 + 0.585i)T - 31iT^{2} \)
37 \( 1 + (1.56 + 1.56i)T + 37iT^{2} \)
41 \( 1 + (-0.966 - 0.966i)T + 41iT^{2} \)
43 \( 1 - 10.3iT - 43T^{2} \)
47 \( 1 + (-4.24 + 4.24i)T - 47iT^{2} \)
53 \( 1 - 13.9iT - 53T^{2} \)
59 \( 1 + (8.81 - 8.81i)T - 59iT^{2} \)
61 \( 1 - 13.1T + 61T^{2} \)
67 \( 1 + (-4.36 + 4.36i)T - 67iT^{2} \)
71 \( 1 + (-0.908 - 0.908i)T + 71iT^{2} \)
73 \( 1 + (11.0 + 11.0i)T + 73iT^{2} \)
79 \( 1 - 6.11T + 79T^{2} \)
83 \( 1 + (-6.49 - 6.49i)T + 83iT^{2} \)
89 \( 1 + (-1.74 + 1.74i)T - 89iT^{2} \)
97 \( 1 + (-5.41 + 5.41i)T - 97iT^{2} \)
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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

−10.86766895310062656209058312399, −9.174958285878960065460213435391, −8.745957999921794892603533243009, −7.904569588393794091338509537864, −7.19355683901351003843285290298, −6.17796506597363154652619235011, −4.79742123270377881958650779418, −3.94446731388682047110086081258, −2.95015511592598679774404067763, −0.884753383857394514679590650690, 2.23029498828409524118879313850, 3.29787920540181946892354421159, 3.97056469357228019911659051120, 4.94710343218536915924963907798, 6.66280298191083178831283655750, 7.18139253132256342540230110112, 8.554303268911564461968872022882, 9.187743505392996868095216841733, 10.28295983616858135914742568115, 11.11661987635983915538659684075

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