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Properties

Label 2-546-21.20-c1-0-23
Degree $2$
Conductor $546$
Sign $0.203 + 0.979i$
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  + i·2-s + (0.0198 + 1.73i)3-s − 4-s − 1.44·5-s + (−1.73 + 0.0198i)6-s + (−2.59 + 0.507i)7-s − i·8-s + (−2.99 + 0.0687i)9-s − 1.44i·10-s − 4.91i·11-s + (−0.0198 − 1.73i)12-s − i·13-s + (−0.507 − 2.59i)14-s + (−0.0287 − 2.50i)15-s + 16-s + 2.13·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.0114 + 0.999i)3-s − 0.5·4-s − 0.647·5-s + (−0.707 + 0.00810i)6-s + (−0.981 + 0.191i)7-s − 0.353i·8-s + (−0.999 + 0.0229i)9-s − 0.457i·10-s − 1.48i·11-s + (−0.00573 − 0.499i)12-s − 0.277i·13-s + (−0.135 − 0.693i)14-s + (−0.00742 − 0.647i)15-s + 0.250·16-s + 0.518·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.203 + 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.203 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.203 + 0.979i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.203 + 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0720307 - 0.0586163i\)
\(L(\frac12)\) \(\approx\) \(0.0720307 - 0.0586163i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (-0.0198 - 1.73i)T \)
7 \( 1 + (2.59 - 0.507i)T \)
13 \( 1 + iT \)
good5 \( 1 + 1.44T + 5T^{2} \)
11 \( 1 + 4.91iT - 11T^{2} \)
17 \( 1 - 2.13T + 17T^{2} \)
19 \( 1 - 2.79iT - 19T^{2} \)
23 \( 1 + 2.25iT - 23T^{2} \)
29 \( 1 + 1.23iT - 29T^{2} \)
31 \( 1 - 5.68iT - 31T^{2} \)
37 \( 1 + 10.3T + 37T^{2} \)
41 \( 1 + 7.01T + 41T^{2} \)
43 \( 1 + 0.729T + 43T^{2} \)
47 \( 1 + 11.8T + 47T^{2} \)
53 \( 1 + 12.9iT - 53T^{2} \)
59 \( 1 - 9.36T + 59T^{2} \)
61 \( 1 - 4.45iT - 61T^{2} \)
67 \( 1 - 1.73T + 67T^{2} \)
71 \( 1 - 7.37iT - 71T^{2} \)
73 \( 1 + 11.4iT - 73T^{2} \)
79 \( 1 - 3.64T + 79T^{2} \)
83 \( 1 + 15.0T + 83T^{2} \)
89 \( 1 + 12.8T + 89T^{2} \)
97 \( 1 + 12.6iT - 97T^{2} \)
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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

−10.35390362887069776932935941515, −9.774459206407570979850586040085, −8.537749279879624675289225887850, −8.324990400868767089236414722541, −6.85658006166742450630273141643, −5.89892736384880418815738886851, −5.14711486350140576381535593259, −3.69478718206035009548574834954, −3.27126868138405733005920466877, −0.05235845827359272972871033263, 1.73033846222393943638746547205, 3.00105090833930991115218518007, 4.07653398708554872489134529219, 5.38408789705573953102503810454, 6.73118557228891930389891243522, 7.32119607263019111161320476326, 8.290114911040738553029598626968, 9.400684442753295427862343752996, 10.05843014994772098416518516106, 11.23476629571164995240880530497

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