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Properties

Label 2-546-91.45-c1-0-6
Degree $2$
Conductor $546$
Sign $0.612 - 0.790i$
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 + (−0.391 + 1.45i)5-s + (−0.258 + 0.965i)6-s + (0.826 − 2.51i)7-s + (0.707 + 0.707i)8-s + (0.499 − 0.866i)9-s + (−0.755 − 1.30i)10-s + (−1.61 + 6.01i)11-s + (−0.500 − 0.866i)12-s + (−2.30 + 2.77i)13-s + (1.19 + 2.36i)14-s + (0.391 + 1.45i)15-s − 1.00·16-s + 7.50·17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.499 − 0.288i)3-s − 0.500i·4-s + (−0.174 + 0.652i)5-s + (−0.105 + 0.394i)6-s + (0.312 − 0.949i)7-s + (0.250 + 0.250i)8-s + (0.166 − 0.288i)9-s + (−0.238 − 0.413i)10-s + (−0.485 + 1.81i)11-s + (−0.144 − 0.250i)12-s + (−0.638 + 0.769i)13-s + (0.318 + 0.631i)14-s + (0.101 + 0.376i)15-s − 0.250·16-s + 1.82·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 - 0.790i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.612 - 0.790i$
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: \((2,\ 546,\ (\ :1/2),\ 0.612 - 0.790i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19847 + 0.587129i\)
\(L(\frac12)\) \(\approx\) \(1.19847 + 0.587129i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.826 + 2.51i)T \)
13 \( 1 + (2.30 - 2.77i)T \)
good5 \( 1 + (0.391 - 1.45i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (1.61 - 6.01i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 - 7.50T + 17T^{2} \)
19 \( 1 + (-7.46 + 2.00i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + 0.0604iT - 23T^{2} \)
29 \( 1 + (-1.60 + 2.78i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (7.86 - 2.10i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-3.41 - 3.41i)T + 37iT^{2} \)
41 \( 1 + (-3.84 + 1.03i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-3.84 + 2.22i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.25 - 0.335i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.81 - 3.14i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (7.03 - 7.03i)T - 59iT^{2} \)
61 \( 1 + (-0.718 - 0.414i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.840 - 0.225i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (0.497 + 0.133i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-2.10 - 7.87i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (5.73 + 9.93i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.74 + 4.74i)T + 83iT^{2} \)
89 \( 1 + (2.02 - 2.02i)T - 89iT^{2} \)
97 \( 1 + (-0.663 + 2.47i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.64487214948655656428269166255, −9.845268410057555912049434196671, −9.375702323012261057907832102738, −7.81134962673909312641626199265, −7.34261121032783639925503932526, −7.02333726377004412906324396475, −5.38772311650313796711702624729, −4.33464955786286071605882681457, −2.92631768883407786970512425277, −1.44659143349559597691639109317, 1.02014365986794088175856267713, 2.81871435464318858125903385262, 3.47160794873557193222591211508, 5.21530494829108956298203823226, 5.67817295366002229146701876097, 7.78918247561400335675915660612, 7.990441601171593920636543937931, 9.008537970014036162546336203777, 9.620892959159855000321340085956, 10.63346623945247218351857235555

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