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Properties

Label 2-546-91.34-c1-0-3
Degree $2$
Conductor $546$
Sign $0.439 - 0.898i$
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 − i·3-s + 1.00i·4-s + (0.0951 − 0.0951i)5-s + (−0.707 + 0.707i)6-s + (−2.64 + 0.0951i)7-s + (0.707 − 0.707i)8-s − 9-s − 0.134·10-s + (−3.64 + 3.64i)11-s + 1.00·12-s + (2 + 3i)13-s + (1.93 + 1.80i)14-s + (−0.0951 − 0.0951i)15-s − 1.00·16-s + 5.98·17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s − 0.577i·3-s + 0.500i·4-s + (0.0425 − 0.0425i)5-s + (−0.288 + 0.288i)6-s + (−0.999 + 0.0359i)7-s + (0.250 − 0.250i)8-s − 0.333·9-s − 0.0425·10-s + (−1.09 + 1.09i)11-s + 0.288·12-s + (0.554 + 0.832i)13-s + (0.517 + 0.481i)14-s + (−0.0245 − 0.0245i)15-s − 0.250·16-s + 1.45·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.439 - 0.898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.439 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.439 - 0.898i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.439 - 0.898i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.480072 + 0.299441i\)
\(L(\frac12)\) \(\approx\) \(0.480072 + 0.299441i\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
7 \( 1 + (2.64 - 0.0951i)T \)
13 \( 1 + (-2 - 3i)T \)
good5 \( 1 + (-0.0951 + 0.0951i)T - 5iT^{2} \)
11 \( 1 + (3.64 - 3.64i)T - 11iT^{2} \)
17 \( 1 - 5.98T + 17T^{2} \)
19 \( 1 + (4.19 - 4.19i)T - 19iT^{2} \)
23 \( 1 + 4.69iT - 23T^{2} \)
29 \( 1 + 4.59T + 29T^{2} \)
31 \( 1 + (0.739 - 0.739i)T - 31iT^{2} \)
37 \( 1 + (4.83 - 4.83i)T - 37iT^{2} \)
41 \( 1 + (-3.04 + 3.04i)T - 41iT^{2} \)
43 \( 1 - 8.78iT - 43T^{2} \)
47 \( 1 + (-3.28 - 3.28i)T + 47iT^{2} \)
53 \( 1 - 1.09T + 53T^{2} \)
59 \( 1 + (-1.30 - 1.30i)T + 59iT^{2} \)
61 \( 1 + 5.98iT - 61T^{2} \)
67 \( 1 + (0.0454 + 0.0454i)T + 67iT^{2} \)
71 \( 1 + (-8.69 - 8.69i)T + 71iT^{2} \)
73 \( 1 + (4.83 + 4.83i)T + 73iT^{2} \)
79 \( 1 + 11.5T + 79T^{2} \)
83 \( 1 + (1.28 - 1.28i)T - 83iT^{2} \)
89 \( 1 + (9.21 + 9.21i)T + 89iT^{2} \)
97 \( 1 + (9.97 - 9.97i)T - 97iT^{2} \)
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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

−10.79641796048675308483463389103, −10.06804347782774898941356278541, −9.346909825767619474272343923598, −8.293629181559284350577494374641, −7.46642748428903467184278949125, −6.60015728498020513508477582879, −5.52212082986451023824476662560, −4.02796505324098686893223794570, −2.84066302175862214618169806178, −1.63046299330269929145389754242, 0.37874954460619744472812084669, 2.79998480808788510229249804935, 3.78388297919481536906324284479, 5.49894473490315007676756543376, 5.80862339336768419204555853580, 7.10685851862953405621548200309, 8.112800109560833821757075958398, 8.832988805031795362110549678206, 9.807993288408236764654697319965, 10.49520603347889958910027707786

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