Properties

Label 2-546-7.4-c3-0-0
Degree $2$
Conductor $546$
Sign $-0.404 - 0.914i$
Analytic cond. $32.2150$
Root an. cond. $5.67582$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − 1.73i)2-s + (1.5 − 2.59i)3-s + (−1.99 + 3.46i)4-s + (3.43 + 5.95i)5-s − 6·6-s + (0.809 − 18.5i)7-s + 7.99·8-s + (−4.5 − 7.79i)9-s + (6.87 − 11.9i)10-s + (−18.6 + 32.3i)11-s + (6.00 + 10.3i)12-s + 13·13-s + (−32.8 + 17.1i)14-s + 20.6·15-s + (−8 − 13.8i)16-s + (−47.4 + 82.1i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.307 + 0.532i)5-s − 0.408·6-s + (0.0437 − 0.999i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.217 − 0.376i)10-s + (−0.512 + 0.886i)11-s + (0.144 + 0.249i)12-s + 0.277·13-s + (−0.627 + 0.326i)14-s + 0.354·15-s + (−0.125 − 0.216i)16-s + (−0.676 + 1.17i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.404 - 0.914i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.404 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.404 - 0.914i$
Analytic conductor: \(32.2150\)
Root analytic conductor: \(5.67582\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :3/2),\ -0.404 - 0.914i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.08381610005\)
\(L(\frac12)\) \(\approx\) \(0.08381610005\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + 1.73i)T \)
3 \( 1 + (-1.5 + 2.59i)T \)
7 \( 1 + (-0.809 + 18.5i)T \)
13 \( 1 - 13T \)
good5 \( 1 + (-3.43 - 5.95i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (18.6 - 32.3i)T + (-665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (47.4 - 82.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (52.7 + 91.3i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (18.5 + 32.1i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 57.9T + 2.43e4T^{2} \)
31 \( 1 + (111. - 193. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-13.7 - 23.8i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 137.T + 6.89e4T^{2} \)
43 \( 1 + 495.T + 7.95e4T^{2} \)
47 \( 1 + (232. + 402. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (136. - 236. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (144. - 249. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (99.3 + 172. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (103. - 179. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 141.T + 3.57e5T^{2} \)
73 \( 1 + (162. - 281. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-529. - 917. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 849.T + 5.71e5T^{2} \)
89 \( 1 + (-692. - 1.19e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 150.T + 9.12e5T^{2} \)
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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

−10.58463209205563521926785461910, −10.05913700683047253232265793273, −8.886979602594535820839765917545, −8.094172954605367139692162929939, −7.05632132405676577197226228694, −6.50876875996685753366836484407, −4.81297090560770657818078459884, −3.76443068629937468415470493447, −2.52777409527052387920842458033, −1.55151364233386196288286391606, 0.02584923422611186985743498030, 1.86181347333188988082796800062, 3.23387360586726140856530069563, 4.69332465179365634390097575260, 5.54400052877536735166770770749, 6.23752279434939814634398025677, 7.68861633853041971601697976280, 8.473292587077232752978640136936, 9.135796894926460834127920286850, 9.757433917905493803371085946403

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