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Properties

Label 2-546-7.4-c3-0-26
Degree $2$
Conductor $546$
Sign $-0.162 + 0.986i$
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 + (−0.436 − 0.756i)5-s − 6·6-s + (−10.8 + 15.0i)7-s + 7.99·8-s + (−4.5 − 7.79i)9-s + (−0.872 + 1.51i)10-s + (0.682 − 1.18i)11-s + (6.00 + 10.3i)12-s + 13·13-s + (36.8 + 3.68i)14-s − 2.61·15-s + (−8 − 13.8i)16-s + (−12.5 + 21.7i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.0390 − 0.0676i)5-s − 0.408·6-s + (−0.583 + 0.812i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.0276 + 0.0478i)10-s + (0.0187 − 0.0324i)11-s + (0.144 + 0.249i)12-s + 0.277·13-s + (0.703 + 0.0703i)14-s − 0.0450·15-s + (−0.125 − 0.216i)16-s + (−0.179 + 0.310i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.162 + 0.986i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.162 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.162 + 0.986i$
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: \((2,\ 546,\ (\ :3/2),\ -0.162 + 0.986i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.405191850\)
\(L(\frac12)\) \(\approx\) \(1.405191850\)
\(L(\frac{5}{2})\) 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 \( 1 + (10.8 - 15.0i)T \)
13 \( 1 - 13T \)
good5 \( 1 + (0.436 + 0.756i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-0.682 + 1.18i)T + (-665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (12.5 - 21.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (6.26 + 10.8i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-39.5 - 68.4i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 3.98T + 2.43e4T^{2} \)
31 \( 1 + (-97.5 + 168. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (90.7 + 157. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 408.T + 6.89e4T^{2} \)
43 \( 1 - 295.T + 7.95e4T^{2} \)
47 \( 1 + (178. + 308. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-181. + 314. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (155. - 269. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-121. - 210. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-128. + 222. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 75.4T + 3.57e5T^{2} \)
73 \( 1 + (-127. + 221. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (27.8 + 48.2i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 33.5T + 5.71e5T^{2} \)
89 \( 1 + (551. + 954. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.33e3T + 9.12e5T^{2} \)
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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

−10.07170109614509293270217168625, −9.181616230951843815969457151918, −8.592569714862271475051351864291, −7.63895585955155448924392158714, −6.55537984449411162856489134116, −5.60827157513150352206445087079, −4.14512384704835108592570632680, −2.97546712142729630976411463028, −2.06140689327859446780712541897, −0.57928545840376412565383915652, 0.996986323588825374815426125722, 2.88761711938818788871091813228, 4.04503673473321020757465147777, 5.00794801752937970532170675883, 6.25271310595312942006673861860, 7.06160685122303986387054534099, 7.950291712212466303708093993510, 8.954486016476252462397557132593, 9.609181128078759009324877207299, 10.53768513450796724013311336627

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