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Properties

Label 2-546-273.200-c1-0-16
Degree $2$
Conductor $546$
Sign $0.962 + 0.272i$
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  + (0.258 − 0.965i)2-s + (1.64 − 0.537i)3-s + (−0.866 − 0.499i)4-s + (−0.868 + 3.24i)5-s + (−0.0931 − 1.72i)6-s + (2.32 − 1.25i)7-s + (−0.707 + 0.707i)8-s + (2.42 − 1.77i)9-s + (2.90 + 1.67i)10-s + (1.63 + 6.11i)11-s + (−1.69 − 0.357i)12-s + (−0.0266 + 3.60i)13-s + (−0.608 − 2.57i)14-s + (0.312 + 5.80i)15-s + (0.500 + 0.866i)16-s + (2.03 − 3.52i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (0.950 − 0.310i)3-s + (−0.433 − 0.249i)4-s + (−0.388 + 1.44i)5-s + (−0.0380 − 0.706i)6-s + (0.880 − 0.474i)7-s + (−0.249 + 0.249i)8-s + (0.807 − 0.590i)9-s + (0.918 + 0.530i)10-s + (0.493 + 1.84i)11-s + (−0.489 − 0.103i)12-s + (−0.00740 + 0.999i)13-s + (−0.162 − 0.688i)14-s + (0.0806 + 1.49i)15-s + (0.125 + 0.216i)16-s + (0.493 − 0.854i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.272i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.962 + 0.272i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (473, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.962 + 0.272i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.13718 - 0.297033i\)
\(L(\frac12)\) \(\approx\) \(2.13718 - 0.297033i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
3 \( 1 + (-1.64 + 0.537i)T \)
7 \( 1 + (-2.32 + 1.25i)T \)
13 \( 1 + (0.0266 - 3.60i)T \)
good5 \( 1 + (0.868 - 3.24i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.63 - 6.11i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-2.03 + 3.52i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.23 + 0.600i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (2.74 + 4.74i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.127iT - 29T^{2} \)
31 \( 1 + (0.443 + 1.65i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-2.03 + 7.58i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-7.12 - 7.12i)T + 41iT^{2} \)
43 \( 1 + 2.56iT - 43T^{2} \)
47 \( 1 + (9.48 + 2.54i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.68 + 1.55i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.65 - 6.18i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.79 - 3.10i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.845 + 3.15i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (8.71 + 8.71i)T + 71iT^{2} \)
73 \( 1 + (4.08 - 1.09i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-3.01 - 5.22i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.395 + 0.395i)T + 83iT^{2} \)
89 \( 1 + (-3.12 - 0.836i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-2.43 + 2.43i)T - 97iT^{2} \)
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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

−10.75488421699267618855567309620, −9.941979396546339960349402139378, −9.232607351824943768100648931786, −7.931427971512318298658331274308, −7.22746810816607538535081050611, −6.59222139748204057923293178088, −4.52418116581268404295692438198, −3.99282418416243247022574969916, −2.62610798762790825588859320353, −1.82153651419469092222579170315, 1.30268398073347246756306907753, 3.29187730412917353379177786449, 4.22232990456890788351455336409, 5.24655776500784231937112883484, 5.97726148069012818406589506878, 7.79759117690738073075072654210, 8.297460499644150082245496413584, 8.626138288602762047740488280291, 9.552133496716263427058773002348, 10.82259859040783543580801225003

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