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Properties

Label 2-546-13.3-c3-0-18
Degree $2$
Conductor $546$
Sign $0.859 + 0.511i$
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 + 7·5-s + (−3 + 5.19i)6-s + (3.5 − 6.06i)7-s + 7.99·8-s + (−4.5 + 7.79i)9-s + (−7 − 12.1i)10-s + (8 + 13.8i)11-s + 12·12-s + (45.5 − 11.2i)13-s − 14·14-s + (−10.5 − 18.1i)15-s + (−8 − 13.8i)16-s + (16.5 − 28.5i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + 0.626·5-s + (−0.204 + 0.353i)6-s + (0.188 − 0.327i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.221 − 0.383i)10-s + (0.219 + 0.379i)11-s + 0.288·12-s + (0.970 − 0.240i)13-s − 0.267·14-s + (−0.180 − 0.313i)15-s + (−0.125 − 0.216i)16-s + (0.235 − 0.407i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.859 + 0.511i$
Analytic conductor: \(32.2150\)
Root analytic conductor: \(5.67582\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :3/2),\ 0.859 + 0.511i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.657306792\)
\(L(\frac12)\) \(\approx\) \(1.657306792\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.5 + 6.06i)T \)
13 \( 1 + (-45.5 + 11.2i)T \)
good5 \( 1 - 7T + 125T^{2} \)
11 \( 1 + (-8 - 13.8i)T + (-665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (-16.5 + 28.5i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (72 - 124. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-22 - 38.1i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-92.5 - 160. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 184T + 2.97e4T^{2} \)
37 \( 1 + (-112.5 - 194. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-82.5 - 142. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-10 + 17.3i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 88T + 1.03e5T^{2} \)
53 \( 1 - 111T + 1.48e5T^{2} \)
59 \( 1 + (110 - 190. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-134.5 + 232. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (350 + 606. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (508 - 879. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 947T + 3.89e5T^{2} \)
79 \( 1 - 1.24e3T + 4.93e5T^{2} \)
83 \( 1 - 1.14e3T + 5.71e5T^{2} \)
89 \( 1 + (429 + 743. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-431 + 746. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−10.32831874177632775759983687995, −9.664783565979109293072630849973, −8.505357431905524333757213264903, −7.82984777783604758128515695779, −6.63819955595262935515369786913, −5.82121769276327689602871501259, −4.55608632008029551678972361576, −3.32184788922733943231276359733, −1.91311040720045177390579711415, −1.04142365834823197438872087429, 0.73986929017033639339357884171, 2.34263532179906412086196596470, 3.98418452603994475671837232896, 5.00508627888155321034310114847, 6.10284160802528264449815421169, 6.48244244290800971753884589590, 7.949197664810170741502295954236, 8.824507694262251561154497301593, 9.396291977999296529396676013556, 10.43837775708358076768877769786

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