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Properties

Label 2-546-91.20-c1-0-8
Degree $2$
Conductor $546$
Sign $0.478 - 0.877i$
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 + (0.866 + 0.5i)3-s + (0.866 − 0.499i)4-s + (−3.01 + 3.01i)5-s + (0.965 + 0.258i)6-s + (2.02 − 1.70i)7-s + (0.707 − 0.707i)8-s + (0.499 + 0.866i)9-s + (−2.13 + 3.69i)10-s + (1.34 + 5.03i)11-s + 12-s + (−2.96 + 2.05i)13-s + (1.51 − 2.17i)14-s + (−4.11 + 1.10i)15-s + (0.500 − 0.866i)16-s + (0.0243 + 0.0422i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.499 + 0.288i)3-s + (0.433 − 0.249i)4-s + (−1.34 + 1.34i)5-s + (0.394 + 0.105i)6-s + (0.764 − 0.644i)7-s + (0.249 − 0.249i)8-s + (0.166 + 0.288i)9-s + (−0.673 + 1.16i)10-s + (0.406 + 1.51i)11-s + 0.288·12-s + (−0.821 + 0.570i)13-s + (0.404 − 0.580i)14-s + (−1.06 + 0.284i)15-s + (0.125 − 0.216i)16-s + (0.00591 + 0.0102i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.478 - 0.877i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.478 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.478 - 0.877i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (475, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.478 - 0.877i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85259 + 1.09996i\)
\(L(\frac12)\) \(\approx\) \(1.85259 + 1.09996i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (-2.02 + 1.70i)T \)
13 \( 1 + (2.96 - 2.05i)T \)
good5 \( 1 + (3.01 - 3.01i)T - 5iT^{2} \)
11 \( 1 + (-1.34 - 5.03i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.0243 - 0.0422i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-7.20 - 1.93i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (2.01 + 1.16i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.957 - 1.65i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.0635 - 0.0635i)T - 31iT^{2} \)
37 \( 1 + (-0.134 - 0.500i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (2.12 + 7.94i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-1.94 + 1.12i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-7.47 - 7.47i)T + 47iT^{2} \)
53 \( 1 + 6.42T + 53T^{2} \)
59 \( 1 + (-3.02 + 11.2i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (0.783 - 0.452i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.54 - 1.48i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-4.13 + 15.4i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (4.68 + 4.68i)T + 73iT^{2} \)
79 \( 1 - 10.1T + 79T^{2} \)
83 \( 1 + (-4.50 + 4.50i)T - 83iT^{2} \)
89 \( 1 + (-7.26 + 1.94i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-4.37 - 1.17i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.97736083571520974614424741547, −10.31592770955202120354694905830, −9.426248897510289545863420419187, −7.71410270340677805388456064700, −7.49944260307942518818158708202, −6.69813388664610283967189356008, −4.89169263795432922292717040178, −4.15269105698514933463671704560, −3.36370528943573610514493208212, −2.08060485865560460572110700447, 1.04574294356982971162886254423, 2.94857988972290324169479797831, 3.95577572289210293389849094435, 5.02709093100431987317790282757, 5.69180617080945643521593126830, 7.33981893907349830343021349674, 8.001668836045156968138463640197, 8.576832646052776535345428429091, 9.425919192246707697039393889438, 11.16866065272188235475091437781

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