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Properties

Label 2-546-273.68-c1-0-5
Degree $2$
Conductor $546$
Sign $0.971 - 0.237i$
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 + (−1.45 − 0.939i)3-s − 4-s + (−0.886 + 1.53i)5-s + (−0.939 + 1.45i)6-s + (−1.67 − 2.04i)7-s + i·8-s + (1.23 + 2.73i)9-s + (1.53 + 0.886i)10-s + (−0.773 − 0.446i)11-s + (1.45 + 0.939i)12-s + (−1.84 + 3.09i)13-s + (−2.04 + 1.67i)14-s + (2.73 − 1.40i)15-s + 16-s + 7.49·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.839 − 0.542i)3-s − 0.5·4-s + (−0.396 + 0.686i)5-s + (−0.383 + 0.593i)6-s + (−0.632 − 0.774i)7-s + 0.353i·8-s + (0.411 + 0.911i)9-s + (0.485 + 0.280i)10-s + (−0.233 − 0.134i)11-s + (0.419 + 0.271i)12-s + (−0.511 + 0.859i)13-s + (−0.547 + 0.447i)14-s + (0.705 − 0.361i)15-s + 0.250·16-s + 1.81·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.237i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 - 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.971 - 0.237i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.971 - 0.237i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.670234 + 0.0806202i\)
\(L(\frac12)\) \(\approx\) \(0.670234 + 0.0806202i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.45 + 0.939i)T \)
7 \( 1 + (1.67 + 2.04i)T \)
13 \( 1 + (1.84 - 3.09i)T \)
good5 \( 1 + (0.886 - 1.53i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.773 + 0.446i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 - 7.49T + 17T^{2} \)
19 \( 1 + (0.697 - 0.402i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 5.45iT - 23T^{2} \)
29 \( 1 + (-7.62 + 4.40i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (8.20 - 4.73i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.69T + 37T^{2} \)
41 \( 1 + (-3.47 - 6.01i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.93 + 3.35i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.19 - 7.25i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.72 + 1.57i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 3.34T + 59T^{2} \)
61 \( 1 + (7.11 - 4.10i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.99 + 6.91i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.97 + 1.14i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (7.54 - 4.35i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.40 - 11.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 13.8T + 83T^{2} \)
89 \( 1 + 15.5T + 89T^{2} \)
97 \( 1 + (-4.24 - 2.45i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.91378503575912685259235704650, −10.18908199016817398585449564390, −9.477313122145405127707706772739, −7.84288540494742806512723313057, −7.29399067675094554635514363621, −6.31877132569066409188861481011, −5.21519128278264771249356691663, −3.98139070466110813155166482414, −2.91886168523866949955146214673, −1.23369131769144991032515777758, 0.50748362732183569479600548178, 3.13624724399638671497496618872, 4.43546591352961749787466717696, 5.35220148525881039289742156955, 5.91155740169208711431098446496, 7.06088938710880712926988438866, 8.119306342677429411517307684706, 8.981909020246237919210713921934, 9.897997703996087540965897065009, 10.51403554815788172972568122952

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