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Properties

Label 2-546-273.257-c1-0-10
Degree $2$
Conductor $546$
Sign $0.697 - 0.716i$
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  − 2-s + (−0.0248 + 1.73i)3-s + 4-s + (−0.511 − 0.295i)5-s + (0.0248 − 1.73i)6-s + (2.62 + 0.304i)7-s − 8-s + (−2.99 − 0.0859i)9-s + (0.511 + 0.295i)10-s + (3.05 − 5.29i)11-s + (−0.0248 + 1.73i)12-s + (1.86 + 3.08i)13-s + (−2.62 − 0.304i)14-s + (0.523 − 0.877i)15-s + 16-s + 7.64·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.0143 + 0.999i)3-s + 0.5·4-s + (−0.228 − 0.131i)5-s + (0.0101 − 0.707i)6-s + (0.993 + 0.115i)7-s − 0.353·8-s + (−0.999 − 0.0286i)9-s + (0.161 + 0.0933i)10-s + (0.921 − 1.59i)11-s + (−0.00716 + 0.499i)12-s + (0.518 + 0.855i)13-s + (−0.702 − 0.0814i)14-s + (0.135 − 0.226i)15-s + 0.250·16-s + 1.85·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.697 - 0.716i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.697 - 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.697 - 0.716i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.697 - 0.716i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08744 + 0.459166i\)
\(L(\frac12)\) \(\approx\) \(1.08744 + 0.459166i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (0.0248 - 1.73i)T \)
7 \( 1 + (-2.62 - 0.304i)T \)
13 \( 1 + (-1.86 - 3.08i)T \)
good5 \( 1 + (0.511 + 0.295i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.05 + 5.29i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 - 7.64T + 17T^{2} \)
19 \( 1 + (1.97 + 3.42i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 8.38iT - 23T^{2} \)
29 \( 1 + (1.39 - 0.803i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.38 - 2.40i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2.22iT - 37T^{2} \)
41 \( 1 + (-1.36 + 0.787i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.90 - 5.03i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.94 - 2.85i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.30 - 1.90i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 4.48iT - 59T^{2} \)
61 \( 1 + (-0.0871 + 0.0502i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.95 - 5.17i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.875 + 1.51i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.41 + 9.37i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.17 - 5.50i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9.07iT - 83T^{2} \)
89 \( 1 - 4.11iT - 89T^{2} \)
97 \( 1 + (2.82 - 4.89i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.02541615144594801239523678105, −9.940020005367527662948851009519, −9.061185091137996817580273096463, −8.504819298935940818402483848163, −7.68447668376271496166136170383, −6.20539009361807337646828574652, −5.41911625754920786402674351718, −4.11413435827784677431115981056, −3.17924103423985828049162744396, −1.25911462916016772748167949418, 1.14733964917080999157038646146, 2.17360968447979776735882076533, 3.79012337188623056375913263096, 5.31199834628491064895812031971, 6.37236489405917687027954272194, 7.39706244454613943400252219312, 7.88511081671241955872291427278, 8.659096163412139730578055440356, 9.864471621919042304665236620244, 10.64948576386246361717415227525

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