Properties

Label 2-546-91.83-c1-0-7
Degree $2$
Conductor $546$
Sign $0.673 + 0.739i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.707i)2-s i·3-s − 1.00i·4-s + (2.56 + 2.56i)5-s + (−0.707 − 0.707i)6-s + (2.56 − 0.648i)7-s + (−0.707 − 0.707i)8-s − 9-s + 3.62·10-s + (−1.64 − 1.64i)11-s − 1.00·12-s + (−2 + 3i)13-s + (1.35 − 2.27i)14-s + (2.56 − 2.56i)15-s − 1.00·16-s + 7.15·17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s − 0.577i·3-s − 0.500i·4-s + (1.14 + 1.14i)5-s + (−0.288 − 0.288i)6-s + (0.969 − 0.245i)7-s + (−0.250 − 0.250i)8-s − 0.333·9-s + 1.14·10-s + (−0.496 − 0.496i)11-s − 0.288·12-s + (−0.554 + 0.832i)13-s + (0.362 − 0.607i)14-s + (0.662 − 0.662i)15-s − 0.250·16-s + 1.73·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.673 + 0.739i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.673 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.673 + 0.739i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (265, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.673 + 0.739i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.16681 - 0.957068i\)
\(L(\frac12)\) \(\approx\) \(2.16681 - 0.957068i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.56 + 0.648i)T \)
13 \( 1 + (2 - 3i)T \)
good5 \( 1 + (-2.56 - 2.56i)T + 5iT^{2} \)
11 \( 1 + (1.64 + 1.64i)T + 11iT^{2} \)
17 \( 1 - 7.15T + 17T^{2} \)
19 \( 1 + (-2.86 - 2.86i)T + 19iT^{2} \)
23 \( 1 + 4.45iT - 23T^{2} \)
29 \( 1 + 9.75T + 29T^{2} \)
31 \( 1 + (3.91 + 3.91i)T + 31iT^{2} \)
37 \( 1 + (-2.48 - 2.48i)T + 37iT^{2} \)
41 \( 1 + (7.53 + 7.53i)T + 41iT^{2} \)
43 \( 1 + 8.62iT - 43T^{2} \)
47 \( 1 + (-0.703 + 0.703i)T - 47iT^{2} \)
53 \( 1 - 2.42T + 53T^{2} \)
59 \( 1 + (10.4 - 10.4i)T - 59iT^{2} \)
61 \( 1 - 7.15iT - 61T^{2} \)
67 \( 1 + (4.53 - 4.53i)T - 67iT^{2} \)
71 \( 1 + (0.456 - 0.456i)T - 71iT^{2} \)
73 \( 1 + (2.48 - 2.48i)T - 73iT^{2} \)
79 \( 1 - 12.0T + 79T^{2} \)
83 \( 1 + (2.70 + 2.70i)T + 83iT^{2} \)
89 \( 1 + (4.75 - 4.75i)T - 89iT^{2} \)
97 \( 1 + (-4.49 - 4.49i)T + 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.64675682391528824210373649178, −10.16215189132078443419113256260, −9.121767277817229951056092702717, −7.72505235466255562968872553869, −7.07240353117700602495452446809, −5.82024135272869919977829631994, −5.37094497495431112454522358779, −3.70161331272321980376207063589, −2.50564617876215992976761105429, −1.60649650450844604077716974414, 1.64881765509968815693724228654, 3.20085298550758547160953035213, 4.82161771366011881936432673618, 5.24786426965867731122706264468, 5.78143168522400171249311067380, 7.54130215563811454406504860745, 8.105754087741952700048220225468, 9.369052704759746362444736148000, 9.698943011633577569351893897663, 10.90112313311448642836358986667

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