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Properties

Label 2-546-13.12-c1-0-7
Degree $2$
Conductor $546$
Sign $0.832 + 0.554i$
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 + 3-s − 4-s + i·5-s − i·6-s + i·7-s + i·8-s + 9-s + 10-s − 3i·11-s − 12-s + (3 + 2i)13-s + 14-s + i·15-s + 16-s + 7·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577·3-s − 0.5·4-s + 0.447i·5-s − 0.408i·6-s + 0.377i·7-s + 0.353i·8-s + 0.333·9-s + 0.316·10-s − 0.904i·11-s − 0.288·12-s + (0.832 + 0.554i)13-s + 0.267·14-s + 0.258i·15-s + 0.250·16-s + 1.69·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.832 + 0.554i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.832 + 0.554i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70292 - 0.515605i\)
\(L(\frac12)\) \(\approx\) \(1.70292 - 0.515605i\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 - iT \)
13 \( 1 + (-3 - 2i)T \)
good5 \( 1 - iT - 5T^{2} \)
11 \( 1 + 3iT - 11T^{2} \)
17 \( 1 - 7T + 17T^{2} \)
19 \( 1 + 3iT - 19T^{2} \)
23 \( 1 - T + 23T^{2} \)
29 \( 1 + T + 29T^{2} \)
31 \( 1 - 8iT - 31T^{2} \)
37 \( 1 - iT - 37T^{2} \)
41 \( 1 + 4iT - 41T^{2} \)
43 \( 1 + 5T + 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 10iT - 59T^{2} \)
61 \( 1 + 13T + 61T^{2} \)
67 \( 1 - 8iT - 67T^{2} \)
71 \( 1 + 6iT - 71T^{2} \)
73 \( 1 - 13iT - 73T^{2} \)
79 \( 1 + 12T + 79T^{2} \)
83 \( 1 + 2iT - 83T^{2} \)
89 \( 1 + 12iT - 89T^{2} \)
97 \( 1 + 6iT - 97T^{2} \)
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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

−10.76478819585551596315768989104, −9.909433120286834848958008835898, −8.912382805672646779752354853771, −8.393621315860024309714270669955, −7.21011486025012968461358392795, −6.08086898664347684779911898670, −4.94946932056961628938788272653, −3.50561592638930415211628589898, −2.95746342164692720974343031467, −1.38390742784676669259892210015, 1.31855408242780957291214467754, 3.23123754426432929052188179294, 4.27516766227616299875620750773, 5.34465390238234539769941603269, 6.35093675030024605547797105182, 7.61178950661489259753692982695, 7.949835094309691188515700603833, 9.037207096057459242685360680499, 9.833323138282610486409897849216, 10.57712839766422400962204337484

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