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Properties

Label 2-546-91.31-c1-0-19
Degree $2$
Conductor $546$
Sign $0.151 + 0.988i$
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 + (0.876 − 3.26i)5-s + (−0.707 − 0.707i)6-s + (1.73 − 1.99i)7-s + (0.707 + 0.707i)8-s + (0.499 + 0.866i)9-s + (1.69 − 2.93i)10-s + (−5.32 + 1.42i)11-s + (−0.5 − 0.866i)12-s + (−0.661 − 3.54i)13-s + (2.19 − 1.48i)14-s + (−2.39 + 2.39i)15-s + (0.500 + 0.866i)16-s + (−1.21 + 2.10i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (−0.499 − 0.288i)3-s + (0.433 + 0.249i)4-s + (0.391 − 1.46i)5-s + (−0.288 − 0.288i)6-s + (0.655 − 0.755i)7-s + (0.249 + 0.249i)8-s + (0.166 + 0.288i)9-s + (0.535 − 0.927i)10-s + (−1.60 + 0.429i)11-s + (−0.144 − 0.249i)12-s + (−0.183 − 0.983i)13-s + (0.585 − 0.396i)14-s + (−0.618 + 0.618i)15-s + (0.125 + 0.216i)16-s + (−0.295 + 0.511i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.151 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.151 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.151 + 0.988i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.151 + 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40502 - 1.20553i\)
\(L(\frac12)\) \(\approx\) \(1.40502 - 1.20553i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 + 1.99i)T \)
13 \( 1 + (0.661 + 3.54i)T \)
good5 \( 1 + (-0.876 + 3.26i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (5.32 - 1.42i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (1.21 - 2.10i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.889 + 3.32i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.824 - 0.476i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 9.96T + 29T^{2} \)
31 \( 1 + (-0.599 + 0.160i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (0.628 - 2.34i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-7.08 - 7.08i)T + 41iT^{2} \)
43 \( 1 + 8.62iT - 43T^{2} \)
47 \( 1 + (0.523 + 0.140i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.63 + 4.56i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.41 - 8.99i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-12.1 + 7.01i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.15 - 11.7i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (8.83 - 8.83i)T - 71iT^{2} \)
73 \( 1 + (0.741 + 2.76i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-0.573 - 0.993i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.46 - 2.46i)T + 83iT^{2} \)
89 \( 1 + (4.42 + 1.18i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (4.97 + 4.97i)T + 97iT^{2} \)
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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

−10.59542062245390213296695609175, −10.02613133261920203947495857480, −8.461483083954463665128606557213, −7.944165533567527335909566108219, −6.93725693607039888231214819624, −5.57441094693080298198424761192, −5.05771144112579531075310702578, −4.35815670866717757824303175471, −2.48488400662328801060799257169, −0.934541738943548646564371675788, 2.24276806883177914619065590228, 2.99665084760245430983196954675, 4.49866020134112609400345890766, 5.47165468203220197852609490418, 6.20893450411453634591885612803, 7.14376828204298052402790666338, 8.197338172727834146598993958556, 9.581759622785122615195289510512, 10.48429446690735690275756102174, 10.96693342702756080166792464962

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