Properties

Label 2-546-91.45-c1-0-14
Degree $2$
Conductor $546$
Sign $-0.0582 + 0.998i$
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 + (0.866 − 0.5i)3-s − 1.00i·4-s + (0.924 − 3.44i)5-s + (−0.258 + 0.965i)6-s + (−2.32 − 1.26i)7-s + (0.707 + 0.707i)8-s + (0.499 − 0.866i)9-s + (1.78 + 3.09i)10-s + (−0.519 + 1.93i)11-s + (−0.500 − 0.866i)12-s + (3.59 − 0.320i)13-s + (2.53 − 0.751i)14-s + (−0.924 − 3.44i)15-s − 1.00·16-s − 4.72·17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.499 − 0.288i)3-s − 0.500i·4-s + (0.413 − 1.54i)5-s + (−0.105 + 0.394i)6-s + (−0.878 − 0.477i)7-s + (0.250 + 0.250i)8-s + (0.166 − 0.288i)9-s + (0.564 + 0.977i)10-s + (−0.156 + 0.584i)11-s + (−0.144 − 0.250i)12-s + (0.996 − 0.0888i)13-s + (0.677 − 0.200i)14-s + (−0.238 − 0.890i)15-s − 0.250·16-s − 1.14·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0582 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0582 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.0582 + 0.998i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.0582 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.769642 - 0.815893i\)
\(L(\frac12)\) \(\approx\) \(0.769642 - 0.815893i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (2.32 + 1.26i)T \)
13 \( 1 + (-3.59 + 0.320i)T \)
good5 \( 1 + (-0.924 + 3.44i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.519 - 1.93i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + 4.72T + 17T^{2} \)
19 \( 1 + (0.658 - 0.176i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + 7.60iT - 23T^{2} \)
29 \( 1 + (-3.67 + 6.36i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (8.58 - 2.30i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-6.47 - 6.47i)T + 37iT^{2} \)
41 \( 1 + (0.562 - 0.150i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (1.13 - 0.652i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (10.1 + 2.73i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-6.72 + 11.6i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.09 + 6.09i)T - 59iT^{2} \)
61 \( 1 + (-8.55 - 4.94i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.25 - 1.67i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-5.57 - 1.49i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.952 - 3.55i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (2.44 + 4.22i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.56 - 7.56i)T + 83iT^{2} \)
89 \( 1 + (10.6 - 10.6i)T - 89iT^{2} \)
97 \( 1 + (-3.35 + 12.5i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.12760150871761081686140886840, −9.566880649459501623484180760790, −8.576244839857607540983254735518, −8.307249946928403226855749644962, −6.88182479270613500431057066796, −6.24475328513358797745697363072, −4.96573701407286588432123632187, −3.99317298892124141515935750455, −2.14657149080685604281065801115, −0.71035589983172260446328976579, 2.11464194087931704755671376903, 3.11559798649171022514753440884, 3.74819047477978065385254738629, 5.71907393920073920715119437263, 6.60124755226686667122598846581, 7.44721552457071709550499637371, 8.709684790185573084168958102945, 9.322792577903283744818678573690, 10.15976993945519196129004722287, 11.01471728316559258520629917629

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