Properties

Label 2-546-39.11-c1-0-12
Degree $2$
Conductor $546$
Sign $0.910 - 0.414i$
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  + (−0.965 − 0.258i)2-s + (1.71 + 0.266i)3-s + (0.866 + 0.499i)4-s + (−0.828 + 0.828i)5-s + (−1.58 − 0.700i)6-s + (0.258 + 0.965i)7-s + (−0.707 − 0.707i)8-s + (2.85 + 0.911i)9-s + (1.01 − 0.585i)10-s + (1.25 − 4.67i)11-s + (1.34 + 1.08i)12-s + (0.0707 + 3.60i)13-s i·14-s + (−1.63 + 1.19i)15-s + (0.500 + 0.866i)16-s + (0.258 − 0.448i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (0.988 + 0.153i)3-s + (0.433 + 0.249i)4-s + (−0.370 + 0.370i)5-s + (−0.646 − 0.285i)6-s + (0.0978 + 0.365i)7-s + (−0.249 − 0.249i)8-s + (0.952 + 0.303i)9-s + (0.320 − 0.185i)10-s + (0.377 − 1.40i)11-s + (0.389 + 0.313i)12-s + (0.0196 + 0.999i)13-s − 0.267i·14-s + (−0.423 + 0.309i)15-s + (0.125 + 0.216i)16-s + (0.0627 − 0.108i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.414i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.910 - 0.414i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.910 - 0.414i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45549 + 0.315916i\)
\(L(\frac12)\) \(\approx\) \(1.45549 + 0.315916i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
3 \( 1 + (-1.71 - 0.266i)T \)
7 \( 1 + (-0.258 - 0.965i)T \)
13 \( 1 + (-0.0707 - 3.60i)T \)
good5 \( 1 + (0.828 - 0.828i)T - 5iT^{2} \)
11 \( 1 + (-1.25 + 4.67i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-0.258 + 0.448i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.50 + 1.47i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-2.54 - 4.40i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.83 - 1.06i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.11 - 4.11i)T + 31iT^{2} \)
37 \( 1 + (9.86 + 2.64i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-6.46 - 1.73i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (0.253 + 0.146i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (8.54 + 8.54i)T + 47iT^{2} \)
53 \( 1 - 9.52iT - 53T^{2} \)
59 \( 1 + (1.53 - 0.411i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-5.90 + 10.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.99 + 11.1i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (0.487 + 1.81i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-4.68 + 4.68i)T - 73iT^{2} \)
79 \( 1 - 2.53T + 79T^{2} \)
83 \( 1 + (11.7 - 11.7i)T - 83iT^{2} \)
89 \( 1 + (-0.717 + 2.67i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-3.95 + 1.05i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.94076351741669427750773998483, −9.659300493412295856011699452029, −9.103937564372702390462149734609, −8.399956984450844984632623789933, −7.45360223400537401654698578391, −6.69537900777489833035526727298, −5.23599756307169174111775846330, −3.64079621861607483495207318387, −3.03724185960524981316623821437, −1.51001566940130424881945615759, 1.16614019255962515743100062280, 2.59005571506184228005735438718, 3.91840097198890822236010613600, 5.01062110525195352880156991134, 6.58597153946674296360163577899, 7.47042515552971744136543152880, 8.027170927816162029063528450322, 8.879481878552499312266489963758, 9.875827653917338348913548532377, 10.24844862279041375079315811037

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