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Properties

Label 2-546-91.74-c1-0-3
Degree $2$
Conductor $546$
Sign $0.531 - 0.847i$
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  − 2-s + (0.5 + 0.866i)3-s + 4-s + (0.623 + 1.07i)5-s + (−0.5 − 0.866i)6-s + (2.30 − 1.30i)7-s − 8-s + (−0.499 + 0.866i)9-s + (−0.623 − 1.07i)10-s + (1.24 + 2.16i)11-s + (0.5 + 0.866i)12-s + (−0.785 + 3.51i)13-s + (−2.30 + 1.30i)14-s + (−0.623 + 1.07i)15-s + 16-s + 0.495·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.288 + 0.499i)3-s + 0.5·4-s + (0.278 + 0.482i)5-s + (−0.204 − 0.353i)6-s + (0.870 − 0.492i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.197 − 0.341i)10-s + (0.376 + 0.651i)11-s + (0.144 + 0.249i)12-s + (−0.217 + 0.976i)13-s + (−0.615 + 0.348i)14-s + (−0.160 + 0.278i)15-s + 0.250·16-s + 0.120·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.531 - 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.531 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.531 - 0.847i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.531 - 0.847i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16698 + 0.645711i\)
\(L(\frac12)\) \(\approx\) \(1.16698 + 0.645711i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-2.30 + 1.30i)T \)
13 \( 1 + (0.785 - 3.51i)T \)
good5 \( 1 + (-0.623 - 1.07i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.24 - 2.16i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 0.495T + 17T^{2} \)
19 \( 1 + (-3.83 + 6.64i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 0.448T + 23T^{2} \)
29 \( 1 + (3.71 - 6.42i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.06 - 7.03i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.74T + 37T^{2} \)
41 \( 1 + (1.87 - 3.24i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.47 - 2.55i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.29 - 5.71i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.86 + 6.69i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 1.45T + 59T^{2} \)
61 \( 1 + (-2.09 + 3.63i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.0138 + 0.0239i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.68 - 8.12i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.07 - 8.78i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.93 + 8.54i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.42T + 83T^{2} \)
89 \( 1 + 11.4T + 89T^{2} \)
97 \( 1 + (0.509 + 0.883i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.95007375416627061594503940990, −9.936526167074786868184604588942, −9.263866084032885806223673701454, −8.476109385214424660716528000531, −7.22397721708549493866983038603, −6.87707848145226368382505700764, −5.24072410215850686712411434384, −4.30510742094948202249394147542, −2.88434261874344011050096735341, −1.58552163656874047479931164612, 1.07907436797511927983477601073, 2.26223663438874164996318449779, 3.68344569906954652173139568837, 5.44078220799943754270902879630, 5.93651942374336153164369823818, 7.51601875914909563732890902418, 7.957919462973531364410093538106, 8.850722620155735332660365593496, 9.563703584780417127159884958416, 10.60248708786142578628056859950

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