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Properties

Label 2-546-13.10-c1-0-7
Degree $2$
Conductor $546$
Sign $0.265 + 0.964i$
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  + (0.866 − 0.5i)2-s + (−0.5 − 0.866i)3-s + (0.499 − 0.866i)4-s + 0.732i·5-s + (−0.866 − 0.499i)6-s + (0.866 + 0.5i)7-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + (0.366 + 0.633i)10-s + (4.5 − 2.59i)11-s − 0.999·12-s + (0.866 − 3.5i)13-s + 0.999·14-s + (0.633 − 0.366i)15-s + (−0.5 − 0.866i)16-s + (−1.13 + 1.96i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.288 − 0.499i)3-s + (0.249 − 0.433i)4-s + 0.327i·5-s + (−0.353 − 0.204i)6-s + (0.327 + 0.188i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.115 + 0.200i)10-s + (1.35 − 0.783i)11-s − 0.288·12-s + (0.240 − 0.970i)13-s + 0.267·14-s + (0.163 − 0.0945i)15-s + (−0.125 − 0.216i)16-s + (−0.275 + 0.476i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.265 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.265 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.265 + 0.964i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.265 + 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58510 - 1.20823i\)
\(L(\frac12)\) \(\approx\) \(1.58510 - 1.20823i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (-0.866 + 3.5i)T \)
good5 \( 1 - 0.732iT - 5T^{2} \)
11 \( 1 + (-4.5 + 2.59i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.13 - 1.96i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.401 + 0.232i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.73 + 6.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.76 - 3.06i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.26iT - 31T^{2} \)
37 \( 1 + (-5.83 + 3.36i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (7.33 - 4.23i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.36 - 7.56i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.92iT - 47T^{2} \)
53 \( 1 + 9.92T + 53T^{2} \)
59 \( 1 + (-7.73 - 4.46i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.86 - 3.23i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.66 + 5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.29 - 3.63i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 1.46iT - 73T^{2} \)
79 \( 1 - 9T + 79T^{2} \)
83 \( 1 - 9.26iT - 83T^{2} \)
89 \( 1 + (14.5 - 8.42i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (12.2 + 7.09i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.98818831520048276307149138032, −9.993870318793570289961264069648, −8.743238234612674527219204522098, −7.967288604788083818557224881256, −6.52543997429274100138818482948, −6.21537209994820167311612232207, −4.97470069741193896304403926685, −3.80530960900091386705203995356, −2.63938249488714978254864761019, −1.14369903608284725194696881675, 1.74365207528250797515571452263, 3.62903464659131082213049936290, 4.41470713949315019862199537618, 5.21206160125379937896494216959, 6.46706840243440355843281942084, 7.06230048433254122061983068154, 8.363042712629729854518786948261, 9.251461091978430513803618732640, 10.00927162009268463229537775197, 11.34316495980674868734778896986

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