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Properties

Label 2-546-39.8-c1-0-10
Degree $2$
Conductor $546$
Sign $0.872 + 0.488i$
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 + (−0.486 − 1.66i)3-s − 1.00i·4-s + (−2.72 + 2.72i)5-s + (1.51 + 0.831i)6-s + (0.707 − 0.707i)7-s + (0.707 + 0.707i)8-s + (−2.52 + 1.61i)9-s − 3.85i·10-s + (0.0114 + 0.0114i)11-s + (−1.66 + 0.486i)12-s + (1.73 − 3.16i)13-s + 1.00i·14-s + (5.86 + 3.20i)15-s − 1.00·16-s + 5.61·17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.280 − 0.959i)3-s − 0.500i·4-s + (−1.22 + 1.22i)5-s + (0.620 + 0.339i)6-s + (0.267 − 0.267i)7-s + (0.250 + 0.250i)8-s + (−0.842 + 0.538i)9-s − 1.22i·10-s + (0.00345 + 0.00345i)11-s + (−0.479 + 0.140i)12-s + (0.480 − 0.877i)13-s + 0.267i·14-s + (1.51 + 0.828i)15-s − 0.250·16-s + 1.36·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.872 + 0.488i$
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: \((2,\ 546,\ (\ :1/2),\ 0.872 + 0.488i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.741369 - 0.193305i\)
\(L(\frac12)\) \(\approx\) \(0.741369 - 0.193305i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.486 + 1.66i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 + (-1.73 + 3.16i)T \)
good5 \( 1 + (2.72 - 2.72i)T - 5iT^{2} \)
11 \( 1 + (-0.0114 - 0.0114i)T + 11iT^{2} \)
17 \( 1 - 5.61T + 17T^{2} \)
19 \( 1 + (0.157 + 0.157i)T + 19iT^{2} \)
23 \( 1 + 2.57T + 23T^{2} \)
29 \( 1 + 7.70iT - 29T^{2} \)
31 \( 1 + (-6.66 - 6.66i)T + 31iT^{2} \)
37 \( 1 + (-7.86 + 7.86i)T - 37iT^{2} \)
41 \( 1 + (-5.18 + 5.18i)T - 41iT^{2} \)
43 \( 1 + 5.09iT - 43T^{2} \)
47 \( 1 + (3.31 + 3.31i)T + 47iT^{2} \)
53 \( 1 + 2.49iT - 53T^{2} \)
59 \( 1 + (-2.10 - 2.10i)T + 59iT^{2} \)
61 \( 1 - 1.82T + 61T^{2} \)
67 \( 1 + (-8.61 - 8.61i)T + 67iT^{2} \)
71 \( 1 + (2.76 - 2.76i)T - 71iT^{2} \)
73 \( 1 + (9.83 - 9.83i)T - 73iT^{2} \)
79 \( 1 + 5.90T + 79T^{2} \)
83 \( 1 + (-11.9 + 11.9i)T - 83iT^{2} \)
89 \( 1 + (-5.15 - 5.15i)T + 89iT^{2} \)
97 \( 1 + (-5.94 - 5.94i)T + 97iT^{2} \)
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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

−10.72927896144246464636423272617, −10.10471659387094785188516340578, −8.435844185012478933214530033916, −7.82140351847505281120071819736, −7.35433724423559964693633618845, −6.41417496604718909266275967266, −5.53905032162210200617148132637, −3.89424212548977646846615550344, −2.65132012189721492051648946123, −0.71401887210069700645622363376, 1.08057655745310155074739966356, 3.21943525194620378797593735903, 4.24376598343848496457095898660, 4.85593010315356511186143256680, 6.14999158747460794271675591912, 7.83081143306894674715565044106, 8.317448613438252595624901443297, 9.224269990090153083725749528915, 9.837948821486449164720549272837, 11.04140415195254076321066783100

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