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Properties

Label 2-546-91.76-c1-0-0
Degree $2$
Conductor $546$
Sign $0.150 + 0.988i$
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 + (−0.866 + 0.5i)3-s + (−0.866 − 0.499i)4-s + (−3.04 + 3.04i)5-s + (−0.258 − 0.965i)6-s + (−1.02 + 2.43i)7-s + (0.707 − 0.707i)8-s + (0.499 − 0.866i)9-s + (−2.15 − 3.72i)10-s + (−2.01 − 0.539i)11-s + 12-s + (0.973 − 3.47i)13-s + (−2.08 − 1.62i)14-s + (1.11 − 4.15i)15-s + (0.500 + 0.866i)16-s + (−0.994 + 1.72i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−0.499 + 0.288i)3-s + (−0.433 − 0.249i)4-s + (−1.36 + 1.36i)5-s + (−0.105 − 0.394i)6-s + (−0.388 + 0.921i)7-s + (0.249 − 0.249i)8-s + (0.166 − 0.288i)9-s + (−0.680 − 1.17i)10-s + (−0.606 − 0.162i)11-s + 0.288·12-s + (0.269 − 0.962i)13-s + (−0.558 − 0.434i)14-s + (0.287 − 1.07i)15-s + (0.125 + 0.216i)16-s + (−0.241 + 0.417i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.150 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.150 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.150 + 0.988i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.150 + 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0911417 - 0.0782842i\)
\(L(\frac12)\) \(\approx\) \(0.0911417 - 0.0782842i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (1.02 - 2.43i)T \)
13 \( 1 + (-0.973 + 3.47i)T \)
good5 \( 1 + (3.04 - 3.04i)T - 5iT^{2} \)
11 \( 1 + (2.01 + 0.539i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.994 - 1.72i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.11 - 4.15i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.911 + 0.526i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.33 - 4.04i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.73 + 5.73i)T - 31iT^{2} \)
37 \( 1 + (-2.17 - 0.582i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (9.88 + 2.64i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (7.68 + 4.43i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.833 + 0.833i)T + 47iT^{2} \)
53 \( 1 + 6.61T + 53T^{2} \)
59 \( 1 + (-7.58 + 2.03i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (9.14 + 5.27i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.51 - 13.1i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-3.38 + 0.908i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-4.18 - 4.18i)T + 73iT^{2} \)
79 \( 1 + 2.85T + 79T^{2} \)
83 \( 1 + (9.59 - 9.59i)T - 83iT^{2} \)
89 \( 1 + (1.02 - 3.81i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-1.29 - 4.83i)T + (-84.0 + 48.5i)T^{2} \)
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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

−11.41244635924600478322705446317, −10.48832618018104624837225308512, −9.984457574396964642868343285455, −8.437085930486302453052606865573, −7.989112928327192305949435860113, −6.88249636735653475961410004222, −6.16798114254954230809024480182, −5.18542742115224681636564934820, −3.80275684497677991090633784587, −2.91549042134991527098845097036, 0.088559566094722090639835191710, 1.25921686894358984149064901107, 3.28327331613262837456743937800, 4.55056188835194569774302168456, 4.82539457252337378321316687251, 6.66393976576948100421144573997, 7.57246415738368906632291046158, 8.370955731367755625093685095553, 9.243461281375022591483644898908, 10.22476327545387284167154001343

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