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Properties

Label 2-546-39.8-c1-0-21
Degree $2$
Conductor $546$
Sign $-0.827 + 0.561i$
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  + (−0.707 + 0.707i)2-s + (1.41 − i)3-s − 1.00i·4-s + (−2.76 + 2.76i)5-s + (−0.292 + 1.70i)6-s + (−0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s + (1.00 − 2.82i)9-s − 3.90i·10-s + (−3.14 − 3.14i)11-s + (−1.00 − 1.41i)12-s + (−3.47 + 0.976i)13-s − 1.00i·14-s + (−1.14 + 6.67i)15-s − 1.00·16-s − 1.90·17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.816 − 0.577i)3-s − 0.500i·4-s + (−1.23 + 1.23i)5-s + (−0.119 + 0.696i)6-s + (−0.267 + 0.267i)7-s + (0.250 + 0.250i)8-s + (0.333 − 0.942i)9-s − 1.23i·10-s + (−0.948 − 0.948i)11-s + (−0.288 − 0.408i)12-s + (−0.962 + 0.270i)13-s − 0.267i·14-s + (−0.295 + 1.72i)15-s − 0.250·16-s − 0.462·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.827 + 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.827 + 0.561i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (281, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.827 + 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0269394 - 0.0877471i\)
\(L(\frac12)\) \(\approx\) \(0.0269394 - 0.0877471i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (-1.41 + i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
13 \( 1 + (3.47 - 0.976i)T \)
good5 \( 1 + (2.76 - 2.76i)T - 5iT^{2} \)
11 \( 1 + (3.14 + 3.14i)T + 11iT^{2} \)
17 \( 1 + 1.90T + 17T^{2} \)
19 \( 1 + (4.76 + 4.76i)T + 19iT^{2} \)
23 \( 1 - 4.56T + 23T^{2} \)
29 \( 1 - 1.03iT - 29T^{2} \)
31 \( 1 + (-0.585 - 0.585i)T + 31iT^{2} \)
37 \( 1 + (3.73 - 3.73i)T - 37iT^{2} \)
41 \( 1 + (4.03 - 4.03i)T - 41iT^{2} \)
43 \( 1 - 3.14iT - 43T^{2} \)
47 \( 1 + (4.24 + 4.24i)T + 47iT^{2} \)
53 \( 1 - 9.60iT - 53T^{2} \)
59 \( 1 + (5.98 + 5.98i)T + 59iT^{2} \)
61 \( 1 + 15.2T + 61T^{2} \)
67 \( 1 + (-8.70 - 8.70i)T + 67iT^{2} \)
71 \( 1 + (-6.49 + 6.49i)T - 71iT^{2} \)
73 \( 1 + (-1.60 + 1.60i)T - 73iT^{2} \)
79 \( 1 + 4.35T + 79T^{2} \)
83 \( 1 + (-0.908 + 0.908i)T - 83iT^{2} \)
89 \( 1 + (-13.0 - 13.0i)T + 89iT^{2} \)
97 \( 1 + (-5.41 - 5.41i)T + 97iT^{2} \)
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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

−10.51273532403153609752525422456, −9.294681439194312407878777398591, −8.427579117421661384118757594418, −7.77796726239773143886343234099, −6.93684616582034278195042901680, −6.44365311699750378073948927803, −4.72299425571691454249178506952, −3.24338362362351414862253981971, −2.53385561774314005273320698917, −0.05262653453102842974306309653, 2.04778183643514603114906425054, 3.43403483004309013326797460585, 4.41850703525903119185572751502, 5.02564874937875874005315545117, 7.23415134867310001570521657696, 7.87398807912465720690385216682, 8.544558113765463783069338697270, 9.356133195784728828469486167294, 10.19783933271150980799559447869, 10.90885751483222537484638842935

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