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Properties

Label 2-546-39.2-c1-0-9
Degree $2$
Conductor $546$
Sign $-0.0886 - 0.996i$
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 + (−1.73 − 0.0700i)3-s + (−0.866 − 0.499i)4-s + (2.88 + 2.88i)5-s + (0.515 − 1.65i)6-s + (0.965 − 0.258i)7-s + (0.707 − 0.707i)8-s + (2.99 + 0.242i)9-s + (−3.53 + 2.04i)10-s + (−0.842 − 0.225i)11-s + (1.46 + 0.925i)12-s + (3.56 − 0.556i)13-s + i·14-s + (−4.79 − 5.19i)15-s + (0.500 + 0.866i)16-s + (1.95 − 3.39i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−0.999 − 0.0404i)3-s + (−0.433 − 0.249i)4-s + (1.29 + 1.29i)5-s + (0.210 − 0.675i)6-s + (0.365 − 0.0978i)7-s + (0.249 − 0.249i)8-s + (0.996 + 0.0808i)9-s + (−1.11 + 0.645i)10-s + (−0.253 − 0.0680i)11-s + (0.422 + 0.267i)12-s + (0.988 − 0.154i)13-s + 0.267i·14-s + (−1.23 − 1.34i)15-s + (0.125 + 0.216i)16-s + (0.474 − 0.822i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0886 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0886 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.0886 - 0.996i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.0886 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.835764 + 0.913437i\)
\(L(\frac12)\) \(\approx\) \(0.835764 + 0.913437i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 + 0.0700i)T \)
7 \( 1 + (-0.965 + 0.258i)T \)
13 \( 1 + (-3.56 + 0.556i)T \)
good5 \( 1 + (-2.88 - 2.88i)T + 5iT^{2} \)
11 \( 1 + (0.842 + 0.225i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.95 + 3.39i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.0297 + 0.111i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-2.17 - 3.76i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.16 + 0.674i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.36 + 1.36i)T - 31iT^{2} \)
37 \( 1 + (1.56 - 5.83i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (2.40 - 8.96i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (5.15 + 2.97i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (8.16 - 8.16i)T - 47iT^{2} \)
53 \( 1 - 0.161iT - 53T^{2} \)
59 \( 1 + (-0.0250 - 0.0933i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-6.92 + 11.9i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-15.2 - 4.08i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-4.81 + 1.28i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (7.81 + 7.81i)T + 73iT^{2} \)
79 \( 1 + 9.72T + 79T^{2} \)
83 \( 1 + (-0.690 - 0.690i)T + 83iT^{2} \)
89 \( 1 + (8.60 + 2.30i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (0.809 + 3.02i)T + (-84.0 + 48.5i)T^{2} \)
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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

−11.03542542350647546532718634390, −10.00641363985226163877127394811, −9.664630264263653424153578823947, −8.166328496643951018599456215949, −7.08742726156508760111955017659, −6.46118430976309068385114588243, −5.71692811034756083677738530246, −4.91923223784557867043595916381, −3.20776302225538119378320259417, −1.49130712598250457962075906305, 1.02656165485024322789344883033, 1.96752631047244113451439207146, 3.99960514751942678372238580272, 5.08593099519321663712152446272, 5.63550330169555531075628703473, 6.67784490873762333831498177615, 8.310905101394581692343687343557, 8.892410394223314144661891039376, 9.967776958264574213042080799321, 10.45168590313491423525844343785

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