Properties

Label 2-546-91.34-c1-0-6
Degree $2$
Conductor $546$
Sign $-0.359 - 0.933i$
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 + (−2.16 + 2.16i)5-s + (0.707 − 0.707i)6-s + (2.62 + 0.292i)7-s + (0.707 − 0.707i)8-s − 9-s + 3.06·10-s + (0.516 − 0.516i)11-s − 1.00·12-s + (3.60 + 0.0469i)13-s + (−1.65 − 2.06i)14-s + (−2.16 − 2.16i)15-s − 1.00·16-s − 2.57·17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.577i·3-s + 0.500i·4-s + (−0.969 + 0.969i)5-s + (0.288 − 0.288i)6-s + (0.993 + 0.110i)7-s + (0.250 − 0.250i)8-s − 0.333·9-s + 0.969·10-s + (0.155 − 0.155i)11-s − 0.288·12-s + (0.999 + 0.0130i)13-s + (−0.441 − 0.552i)14-s + (−0.559 − 0.559i)15-s − 0.250·16-s − 0.624·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.359 - 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.359 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.359 - 0.933i$
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: \((2,\ 546,\ (\ :1/2),\ -0.359 - 0.933i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.448769 + 0.653605i\)
\(L(\frac12)\) \(\approx\) \(0.448769 + 0.653605i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.62 - 0.292i)T \)
13 \( 1 + (-3.60 - 0.0469i)T \)
good5 \( 1 + (2.16 - 2.16i)T - 5iT^{2} \)
11 \( 1 + (-0.516 + 0.516i)T - 11iT^{2} \)
17 \( 1 + 2.57T + 17T^{2} \)
19 \( 1 + (3.82 - 3.82i)T - 19iT^{2} \)
23 \( 1 - 6.97iT - 23T^{2} \)
29 \( 1 + 6.32T + 29T^{2} \)
31 \( 1 + (4.33 - 4.33i)T - 31iT^{2} \)
37 \( 1 + (3.58 - 3.58i)T - 37iT^{2} \)
41 \( 1 + (0.176 - 0.176i)T - 41iT^{2} \)
43 \( 1 + 7.89iT - 43T^{2} \)
47 \( 1 + (-1.41 - 1.41i)T + 47iT^{2} \)
53 \( 1 + 0.514T + 53T^{2} \)
59 \( 1 + (3.17 + 3.17i)T + 59iT^{2} \)
61 \( 1 + 6.41iT - 61T^{2} \)
67 \( 1 + (3.96 + 3.96i)T + 67iT^{2} \)
71 \( 1 + (-8.12 - 8.12i)T + 71iT^{2} \)
73 \( 1 + (-11.6 - 11.6i)T + 73iT^{2} \)
79 \( 1 - 10.9T + 79T^{2} \)
83 \( 1 + (-1.01 + 1.01i)T - 83iT^{2} \)
89 \( 1 + (-1.10 - 1.10i)T + 89iT^{2} \)
97 \( 1 + (9.34 - 9.34i)T - 97iT^{2} \)
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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

−11.04490061116561557794026618902, −10.54480079899871866841962415805, −9.301585264589211097152104418336, −8.434056772785093145979829570609, −7.76329497485876543916616050497, −6.74594565758980879084041606589, −5.40457148306889750099931758824, −3.96732833121641968829494732540, −3.48552596847837310251339022203, −1.85674640290608875164837534144, 0.55118220855173462400731389571, 1.95472897536364406063810010244, 4.08307034644370563219510960333, 4.85452600993166717075466883423, 6.08577856744845749437963509708, 7.13363742136715842307920219558, 7.985271452779885404090175570655, 8.595610067709012154257147863807, 9.143600084395631030553712742806, 10.87140111184602596684161187555

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