Properties

Label 2-546-91.34-c1-0-0
Degree $2$
Conductor $546$
Sign $-0.864 - 0.503i$
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 + (−0.0951 + 0.0951i)5-s + (0.707 − 0.707i)6-s + (−0.0951 + 2.64i)7-s + (0.707 − 0.707i)8-s − 9-s + 0.134·10-s + (−3.64 + 3.64i)11-s − 1.00·12-s + (−2 − 3i)13-s + (1.93 − 1.80i)14-s + (−0.0951 − 0.0951i)15-s − 1.00·16-s − 5.98·17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.577i·3-s + 0.500i·4-s + (−0.0425 + 0.0425i)5-s + (0.288 − 0.288i)6-s + (−0.0359 + 0.999i)7-s + (0.250 − 0.250i)8-s − 0.333·9-s + 0.0425·10-s + (−1.09 + 1.09i)11-s − 0.288·12-s + (−0.554 − 0.832i)13-s + (0.517 − 0.481i)14-s + (−0.0245 − 0.0245i)15-s − 0.250·16-s − 1.45·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.864 - 0.503i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.864 - 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.864 - 0.503i$
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: \((2,\ 546,\ (\ :1/2),\ -0.864 - 0.503i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.106164 + 0.393231i\)
\(L(\frac12)\) \(\approx\) \(0.106164 + 0.393231i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0951 - 2.64i)T \)
13 \( 1 + (2 + 3i)T \)
good5 \( 1 + (0.0951 - 0.0951i)T - 5iT^{2} \)
11 \( 1 + (3.64 - 3.64i)T - 11iT^{2} \)
17 \( 1 + 5.98T + 17T^{2} \)
19 \( 1 + (-4.19 + 4.19i)T - 19iT^{2} \)
23 \( 1 + 4.69iT - 23T^{2} \)
29 \( 1 + 4.59T + 29T^{2} \)
31 \( 1 + (-0.739 + 0.739i)T - 31iT^{2} \)
37 \( 1 + (4.83 - 4.83i)T - 37iT^{2} \)
41 \( 1 + (3.04 - 3.04i)T - 41iT^{2} \)
43 \( 1 - 8.78iT - 43T^{2} \)
47 \( 1 + (3.28 + 3.28i)T + 47iT^{2} \)
53 \( 1 - 1.09T + 53T^{2} \)
59 \( 1 + (1.30 + 1.30i)T + 59iT^{2} \)
61 \( 1 - 5.98iT - 61T^{2} \)
67 \( 1 + (0.0454 + 0.0454i)T + 67iT^{2} \)
71 \( 1 + (-8.69 - 8.69i)T + 71iT^{2} \)
73 \( 1 + (-4.83 - 4.83i)T + 73iT^{2} \)
79 \( 1 + 11.5T + 79T^{2} \)
83 \( 1 + (-1.28 + 1.28i)T - 83iT^{2} \)
89 \( 1 + (-9.21 - 9.21i)T + 89iT^{2} \)
97 \( 1 + (-9.97 + 9.97i)T - 97iT^{2} \)
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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

−11.10233095268026542429704494072, −10.17652912258863932722572432350, −9.528152986973598342440522372011, −8.733692675625309829516145923405, −7.80749448790132785244048560435, −6.80402720937937783962777631325, −5.30043314097502105756708201574, −4.68593743647949807625755054106, −3.04164149823348289147813919029, −2.26252672335382339710729988371, 0.25736904524656647619023009571, 1.96402794558974565065255035754, 3.59749226875040220013926904144, 4.99013545398374012889155285276, 6.02164165475908481713071787597, 7.02931077715164528421084172991, 7.64788298255783038916104359940, 8.489302041439476499159027441428, 9.445597193658361028293290128287, 10.46341929613216769902638781000

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