Properties

Label 2-546-273.257-c1-0-35
Degree $2$
Conductor $546$
Sign $0.0333 + 0.999i$
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 + (1.13 − 1.30i)3-s + 4-s + (−1.26 − 0.730i)5-s + (1.13 − 1.30i)6-s + (−1.08 − 2.41i)7-s + 8-s + (−0.421 − 2.97i)9-s + (−1.26 − 0.730i)10-s + (−1.75 + 3.04i)11-s + (1.13 − 1.30i)12-s + (0.214 − 3.59i)13-s + (−1.08 − 2.41i)14-s + (−2.39 + 0.825i)15-s + 16-s + 7.58·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.655 − 0.755i)3-s + 0.5·4-s + (−0.565 − 0.326i)5-s + (0.463 − 0.533i)6-s + (−0.411 − 0.911i)7-s + 0.353·8-s + (−0.140 − 0.990i)9-s + (−0.399 − 0.230i)10-s + (−0.530 + 0.918i)11-s + (0.327 − 0.377i)12-s + (0.0595 − 0.998i)13-s + (−0.290 − 0.644i)14-s + (−0.617 + 0.213i)15-s + 0.250·16-s + 1.84·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0333 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0333 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.0333 + 0.999i$
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: \((2,\ 546,\ (\ :1/2),\ 0.0333 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66324 - 1.60862i\)
\(L(\frac12)\) \(\approx\) \(1.66324 - 1.60862i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.13 + 1.30i)T \)
7 \( 1 + (1.08 + 2.41i)T \)
13 \( 1 + (-0.214 + 3.59i)T \)
good5 \( 1 + (1.26 + 0.730i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.75 - 3.04i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 - 7.58T + 17T^{2} \)
19 \( 1 + (1.72 + 2.99i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.60iT - 23T^{2} \)
29 \( 1 + (-0.170 + 0.0985i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.34 - 9.25i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 5.56iT - 37T^{2} \)
41 \( 1 + (2.60 - 1.50i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.61 + 9.71i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-11.0 - 6.39i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.21 - 2.43i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 1.30iT - 59T^{2} \)
61 \( 1 + (-0.865 + 0.499i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.78 - 2.76i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.33 - 9.24i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.94 + 5.09i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.174 + 0.302i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.72iT - 83T^{2} \)
89 \( 1 + 14.4iT - 89T^{2} \)
97 \( 1 + (1.72 - 2.99i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.49046243474860293784968019931, −9.908259677827302974240458509395, −8.520633984753779699834888146261, −7.58434138496589870038717510430, −7.25589758934303043468518745465, −6.01702201667707239203986182217, −4.81078952617090790036661607537, −3.66201561873725612883688990246, −2.81954154103993170764954454419, −1.06905567661983405791616363325, 2.39363467409027752458129525815, 3.34187998026111384869989238402, 4.12935929926477188456375317510, 5.42590860462088700461698054501, 6.14948020984493305621591370475, 7.61348758835277308121070049841, 8.271424818594521082110393325007, 9.313079995696410137394373962493, 10.16210960398951777793936389480, 11.10456828709105156889852103299

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