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Properties

Label 2-546-39.8-c1-0-9
Degree $2$
Conductor $546$
Sign $0.980 - 0.194i$
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 + (−1.14 + 1.29i)3-s − 1.00i·4-s + (0.237 − 0.237i)5-s + (0.103 + 1.72i)6-s + (−0.707 + 0.707i)7-s + (−0.707 − 0.707i)8-s + (−0.359 − 2.97i)9-s − 0.335i·10-s + (2.55 + 2.55i)11-s + (1.29 + 1.14i)12-s + (3.37 + 1.25i)13-s + 1.00i·14-s + (0.0348 + 0.579i)15-s − 1.00·16-s + 5.26·17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.663 + 0.748i)3-s − 0.500i·4-s + (0.106 − 0.106i)5-s + (0.0424 + 0.705i)6-s + (−0.267 + 0.267i)7-s + (−0.250 − 0.250i)8-s + (−0.119 − 0.992i)9-s − 0.106i·10-s + (0.770 + 0.770i)11-s + (0.374 + 0.331i)12-s + (0.937 + 0.348i)13-s + 0.267i·14-s + (0.00900 + 0.149i)15-s − 0.250·16-s + 1.27·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.194i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.980 - 0.194i$
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: \((2,\ 546,\ (\ :1/2),\ 0.980 - 0.194i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59857 + 0.156814i\)
\(L(\frac12)\) \(\approx\) \(1.59857 + 0.156814i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.14 - 1.29i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
13 \( 1 + (-3.37 - 1.25i)T \)
good5 \( 1 + (-0.237 + 0.237i)T - 5iT^{2} \)
11 \( 1 + (-2.55 - 2.55i)T + 11iT^{2} \)
17 \( 1 - 5.26T + 17T^{2} \)
19 \( 1 + (-0.906 - 0.906i)T + 19iT^{2} \)
23 \( 1 + 1.48T + 23T^{2} \)
29 \( 1 + 7.36iT - 29T^{2} \)
31 \( 1 + (-7.20 - 7.20i)T + 31iT^{2} \)
37 \( 1 + (2.83 - 2.83i)T - 37iT^{2} \)
41 \( 1 + (-1.32 + 1.32i)T - 41iT^{2} \)
43 \( 1 - 2.75iT - 43T^{2} \)
47 \( 1 + (2.27 + 2.27i)T + 47iT^{2} \)
53 \( 1 + 8.80iT - 53T^{2} \)
59 \( 1 + (0.785 + 0.785i)T + 59iT^{2} \)
61 \( 1 - 8.74T + 61T^{2} \)
67 \( 1 + (9.68 + 9.68i)T + 67iT^{2} \)
71 \( 1 + (-4.94 + 4.94i)T - 71iT^{2} \)
73 \( 1 + (4.00 - 4.00i)T - 73iT^{2} \)
79 \( 1 + 8.50T + 79T^{2} \)
83 \( 1 + (3.05 - 3.05i)T - 83iT^{2} \)
89 \( 1 + (-1.62 - 1.62i)T + 89iT^{2} \)
97 \( 1 + (4.86 + 4.86i)T + 97iT^{2} \)
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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

−10.89764728762223431343138440399, −9.912716159016952821849148305933, −9.544684484645009506060827490974, −8.396621500881133484816939960042, −6.82978188385947369232974970586, −6.02129889582371286769205707141, −5.13827002456702926327949355424, −4.10787355957312896685689347527, −3.25591093242019012030873007593, −1.38271731331638374575734185294, 1.07406411149926278383208740382, 2.98376970013329652976379866795, 4.17610461015540164832671229180, 5.57608163086084874070974800894, 6.11070874338930857040683180528, 6.94694020620384051022175598828, 7.907937309830033620181316146406, 8.695910767795913786755305445172, 10.05904112292115349125150204441, 10.97424531906844250106458392828

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