Properties

Label 2-546-91.23-c1-0-12
Degree $2$
Conductor $546$
Sign $0.132 + 0.991i$
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  + i·2-s + (−0.5 + 0.866i)3-s − 4-s + (0.440 + 0.254i)5-s + (−0.866 − 0.5i)6-s + (−0.212 − 2.63i)7-s i·8-s + (−0.499 − 0.866i)9-s + (−0.254 + 0.440i)10-s + (−3.34 − 1.93i)11-s + (0.5 − 0.866i)12-s + (−3.12 − 1.79i)13-s + (2.63 − 0.212i)14-s + (−0.440 + 0.254i)15-s + 16-s − 7.78·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.288 + 0.499i)3-s − 0.5·4-s + (0.196 + 0.113i)5-s + (−0.353 − 0.204i)6-s + (−0.0804 − 0.996i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + (−0.0804 + 0.139i)10-s + (−1.00 − 0.582i)11-s + (0.144 − 0.249i)12-s + (−0.866 − 0.498i)13-s + (0.704 − 0.0569i)14-s + (−0.113 + 0.0656i)15-s + 0.250·16-s − 1.88·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.132 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.132 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.132 + 0.991i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.132 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.321925 - 0.281772i\)
\(L(\frac12)\) \(\approx\) \(0.321925 - 0.281772i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (0.212 + 2.63i)T \)
13 \( 1 + (3.12 + 1.79i)T \)
good5 \( 1 + (-0.440 - 0.254i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.34 + 1.93i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + 7.78T + 17T^{2} \)
19 \( 1 + (0.0999 - 0.0577i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 1.22T + 23T^{2} \)
29 \( 1 + (-1.86 - 3.23i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.83 + 2.21i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.73iT - 37T^{2} \)
41 \( 1 + (-0.138 + 0.0796i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.153 - 0.266i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.61 - 3.81i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.08 + 3.60i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 8.36iT - 59T^{2} \)
61 \( 1 + (3.07 + 5.32i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.33 + 0.770i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.505 + 0.292i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.78 - 2.76i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.16 + 3.75i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.96iT - 83T^{2} \)
89 \( 1 + 4.53iT - 89T^{2} \)
97 \( 1 + (13.4 + 7.73i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.52479782952262203803552062863, −9.816766287166166319062971549953, −8.757097077122926625672247763945, −7.83462667290507301880437594439, −6.93124582519526469410697948731, −6.03671206333335274500977940990, −4.94663476923147260865452533174, −4.20050846344862161942792073674, −2.74137245216132401106465421354, −0.23793428612221239750616829328, 2.00423005395776057909964555268, 2.67134375026314415690164843161, 4.51171722288743467266007187798, 5.26819543541947347598706741884, 6.39987231459783376059318409547, 7.43492153183466965083982132850, 8.541305304260616706044180104475, 9.304527798085556376194472589139, 10.17589826850256300973057065247, 11.16495395922087530637825173323

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