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Properties

Label 2-546-91.76-c1-0-1
Degree $2$
Conductor $546$
Sign $-0.984 + 0.173i$
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  + (−0.258 + 0.965i)2-s + (0.866 − 0.5i)3-s + (−0.866 − 0.499i)4-s + (−0.301 + 0.301i)5-s + (0.258 + 0.965i)6-s + (−2.51 + 0.831i)7-s + (0.707 − 0.707i)8-s + (0.499 − 0.866i)9-s + (−0.213 − 0.369i)10-s + (−5.62 − 1.50i)11-s − 12-s + (−2.97 + 2.04i)13-s + (−0.152 − 2.64i)14-s + (−0.110 + 0.412i)15-s + (0.500 + 0.866i)16-s + (−3.43 + 5.94i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (0.499 − 0.288i)3-s + (−0.433 − 0.249i)4-s + (−0.134 + 0.134i)5-s + (0.105 + 0.394i)6-s + (−0.949 + 0.314i)7-s + (0.249 − 0.249i)8-s + (0.166 − 0.288i)9-s + (−0.0674 − 0.116i)10-s + (−1.69 − 0.454i)11-s − 0.288·12-s + (−0.824 + 0.566i)13-s + (−0.0407 − 0.705i)14-s + (−0.0285 + 0.106i)15-s + (0.125 + 0.216i)16-s + (−0.832 + 1.44i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.984 + 0.173i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.984 + 0.173i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0260450 - 0.297604i\)
\(L(\frac12)\) \(\approx\) \(0.0260450 - 0.297604i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (2.51 - 0.831i)T \)
13 \( 1 + (2.97 - 2.04i)T \)
good5 \( 1 + (0.301 - 0.301i)T - 5iT^{2} \)
11 \( 1 + (5.62 + 1.50i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (3.43 - 5.94i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.0339 - 0.126i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-4.00 + 2.31i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.99 + 3.45i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.778 - 0.778i)T - 31iT^{2} \)
37 \( 1 + (-3.22 - 0.862i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (10.8 + 2.90i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-5.32 - 3.07i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.709 + 0.709i)T + 47iT^{2} \)
53 \( 1 + 7.17T + 53T^{2} \)
59 \( 1 + (-3.84 + 1.02i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.320 - 0.184i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.57 + 13.3i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (12.5 - 3.35i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-0.569 - 0.569i)T + 73iT^{2} \)
79 \( 1 - 3.25T + 79T^{2} \)
83 \( 1 + (-3.34 + 3.34i)T - 83iT^{2} \)
89 \( 1 + (1.31 - 4.89i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-3.18 - 11.8i)T + (-84.0 + 48.5i)T^{2} \)
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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

−11.05300052601991051842284331922, −10.21781756224791253857974950462, −9.321099619365212990534288706040, −8.516849923621967303228799623727, −7.69905505819572312206630740904, −6.82953692212857967585680343156, −5.96595184355212112576384424276, −4.86252706371007909803787557931, −3.45863404532424976629260474149, −2.29802706924294423678467059539, 0.15771956887212505562275389367, 2.49750061551856953966712236405, 3.10602946982757284924359998282, 4.53876303619048372672377505030, 5.31210477743157387579686867584, 7.04241554007548810854772740202, 7.69256745195302280814590695884, 8.785281331481507117520387506461, 9.676931678013033695343193747422, 10.19196972431347844076469191766

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