Properties

Label 2-546-91.34-c1-0-8
Degree $2$
Conductor $546$
Sign $-0.802 - 0.596i$
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  + (0.707 + 0.707i)2-s + i·3-s + 1.00i·4-s + (−1.15 + 1.15i)5-s + (−0.707 + 0.707i)6-s + (2.02 + 1.70i)7-s + (−0.707 + 0.707i)8-s − 9-s − 1.63·10-s + (1.37 − 1.37i)11-s − 1.00·12-s + (−1.50 + 3.27i)13-s + (0.222 + 2.63i)14-s + (−1.15 − 1.15i)15-s − 1.00·16-s − 1.50·17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.577i·3-s + 0.500i·4-s + (−0.517 + 0.517i)5-s + (−0.288 + 0.288i)6-s + (0.763 + 0.645i)7-s + (−0.250 + 0.250i)8-s − 0.333·9-s − 0.517·10-s + (0.415 − 0.415i)11-s − 0.288·12-s + (−0.416 + 0.909i)13-s + (0.0593 + 0.704i)14-s + (−0.298 − 0.298i)15-s − 0.250·16-s − 0.365·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.802 - 0.596i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.802 - 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.802 - 0.596i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.802 - 0.596i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.520184 + 1.57186i\)
\(L(\frac12)\) \(\approx\) \(0.520184 + 1.57186i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 - iT \)
7 \( 1 + (-2.02 - 1.70i)T \)
13 \( 1 + (1.50 - 3.27i)T \)
good5 \( 1 + (1.15 - 1.15i)T - 5iT^{2} \)
11 \( 1 + (-1.37 + 1.37i)T - 11iT^{2} \)
17 \( 1 + 1.50T + 17T^{2} \)
19 \( 1 + (0.934 - 0.934i)T - 19iT^{2} \)
23 \( 1 + 4.19iT - 23T^{2} \)
29 \( 1 + 0.406T + 29T^{2} \)
31 \( 1 + (2.31 - 2.31i)T - 31iT^{2} \)
37 \( 1 + (-0.257 + 0.257i)T - 37iT^{2} \)
41 \( 1 + (-3.60 + 3.60i)T - 41iT^{2} \)
43 \( 1 - 2.46iT - 43T^{2} \)
47 \( 1 + (1.41 + 1.41i)T + 47iT^{2} \)
53 \( 1 - 13.4T + 53T^{2} \)
59 \( 1 + (-8.75 - 8.75i)T + 59iT^{2} \)
61 \( 1 + 11.7iT - 61T^{2} \)
67 \( 1 + (-11.1 - 11.1i)T + 67iT^{2} \)
71 \( 1 + (5.18 + 5.18i)T + 71iT^{2} \)
73 \( 1 + (2.56 + 2.56i)T + 73iT^{2} \)
79 \( 1 - 1.42T + 79T^{2} \)
83 \( 1 + (-4.90 + 4.90i)T - 83iT^{2} \)
89 \( 1 + (-2.03 - 2.03i)T + 89iT^{2} \)
97 \( 1 + (-10.2 + 10.2i)T - 97iT^{2} \)
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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

−11.38112681774762719652998171795, −10.42606787767373780177415576704, −9.110637407856498803725840511089, −8.557118030816410349627315445273, −7.47266348344192726764469488826, −6.57080029172342849100544506355, −5.51789818415329465658579081182, −4.54277261530459335503779521909, −3.68449735809379613504525882092, −2.33912737336825666681579191485, 0.844475772182072899889591142894, 2.21894489055507114865926331312, 3.74967385998910336813199788179, 4.65709499604539760677481861391, 5.59465141684039342000958097985, 6.91510659285921809499471005921, 7.72972099547139529794540804481, 8.556603814133231842011386715264, 9.711247391724814154539681275802, 10.68622467140158074561747366136

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