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Properties

Label 2-546-13.4-c1-0-6
Degree $2$
Conductor $546$
Sign $0.803 - 0.594i$
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  + (0.866 + 0.5i)2-s + (0.5 − 0.866i)3-s + (0.499 + 0.866i)4-s + 1.78i·5-s + (0.866 − 0.499i)6-s + (0.866 − 0.5i)7-s + 0.999i·8-s + (−0.499 − 0.866i)9-s + (−0.894 + 1.54i)10-s + (2.74 + 1.58i)11-s + 0.999·12-s + (1.47 − 3.28i)13-s + 0.999·14-s + (1.54 + 0.894i)15-s + (−0.5 + 0.866i)16-s + (2.78 + 4.81i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.288 − 0.499i)3-s + (0.249 + 0.433i)4-s + 0.799i·5-s + (0.353 − 0.204i)6-s + (0.327 − 0.188i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + (−0.282 + 0.489i)10-s + (0.828 + 0.478i)11-s + 0.288·12-s + (0.410 − 0.911i)13-s + 0.267·14-s + (0.399 + 0.230i)15-s + (−0.125 + 0.216i)16-s + (0.674 + 1.16i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.803 - 0.594i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.803 - 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.803 - 0.594i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.803 - 0.594i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.25951 + 0.745257i\)
\(L(\frac12)\) \(\approx\) \(2.25951 + 0.745257i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (-1.47 + 3.28i)T \)
good5 \( 1 - 1.78iT - 5T^{2} \)
11 \( 1 + (-2.74 - 1.58i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.78 - 4.81i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.36 - 3.09i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.06 + 5.31i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.03 - 1.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.63iT - 31T^{2} \)
37 \( 1 + (2.68 + 1.54i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.29 + 0.749i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.81 + 8.34i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 10.5iT - 47T^{2} \)
53 \( 1 - 3.60T + 53T^{2} \)
59 \( 1 + (2.40 - 1.39i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.844 + 1.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.0 + 5.77i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.518 - 0.299i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 0.423iT - 73T^{2} \)
79 \( 1 - 6.96T + 79T^{2} \)
83 \( 1 + 4.30iT - 83T^{2} \)
89 \( 1 + (14.1 + 8.14i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (15.1 - 8.74i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.68229191528527728150875375908, −10.43948038058133692589772253647, −8.742029056583244740512454014891, −8.162461478709186409087643117594, −7.01680028345813122810977158556, −6.52420849918109294422129972626, −5.45656909815578690205575662469, −4.06099543942088371588117575318, −3.18616859703353666726230207951, −1.76186345992483143026681131053, 1.37603737750243930769747092059, 2.90985402050510826591347431082, 4.14250299671800697735016151326, 4.81894784126759169057010764486, 5.84951530656746419733850756187, 6.96323507245106491752274989910, 8.281647319434559105955702628398, 9.165286582003996790526038314712, 9.613614043774327106423212575343, 11.07696633691376025357402243802

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