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Properties

Label 2-546-91.41-c1-0-11
Degree $2$
Conductor $546$
Sign $0.371 + 0.928i$
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 + (−1.20 − 1.20i)5-s + (−0.965 + 0.258i)6-s + (2.62 + 0.352i)7-s + (−0.707 − 0.707i)8-s + (0.499 − 0.866i)9-s + (0.852 + 1.47i)10-s + (0.674 − 2.51i)11-s + 12-s + (0.992 + 3.46i)13-s + (−2.44 − 1.01i)14-s + (−1.64 − 0.441i)15-s + (0.500 + 0.866i)16-s + (2.23 − 3.87i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (0.499 − 0.288i)3-s + (0.433 + 0.249i)4-s + (−0.539 − 0.539i)5-s + (−0.394 + 0.105i)6-s + (0.991 + 0.133i)7-s + (−0.249 − 0.249i)8-s + (0.166 − 0.288i)9-s + (0.269 + 0.467i)10-s + (0.203 − 0.759i)11-s + 0.288·12-s + (0.275 + 0.961i)13-s + (−0.652 − 0.272i)14-s + (−0.425 − 0.113i)15-s + (0.125 + 0.216i)16-s + (0.542 − 0.939i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.371 + 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.371 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.371 + 0.928i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.371 + 0.928i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05172 - 0.711583i\)
\(L(\frac12)\) \(\approx\) \(1.05172 - 0.711583i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.62 - 0.352i)T \)
13 \( 1 + (-0.992 - 3.46i)T \)
good5 \( 1 + (1.20 + 1.20i)T + 5iT^{2} \)
11 \( 1 + (-0.674 + 2.51i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-2.23 + 3.87i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.65 - 0.710i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-2.92 + 1.69i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.61 - 2.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.57 + 2.57i)T + 31iT^{2} \)
37 \( 1 + (-2.58 + 9.64i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.0733 + 0.273i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-4.82 - 2.78i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.80 - 3.80i)T - 47iT^{2} \)
53 \( 1 + 3.12T + 53T^{2} \)
59 \( 1 + (2.54 + 9.51i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (1.86 + 1.07i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.32 - 0.623i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.73 - 6.49i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (0.328 - 0.328i)T - 73iT^{2} \)
79 \( 1 - 7.61T + 79T^{2} \)
83 \( 1 + (1.89 + 1.89i)T + 83iT^{2} \)
89 \( 1 + (-1.61 - 0.431i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (3.24 - 0.868i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.81639380895748866565135125995, −9.402600533981882192989865564543, −8.830419060578057819698919378007, −8.110538245618585392433233778313, −7.39637454896866822971889209648, −6.25129082533291360426495734349, −4.85162453743216835670838292334, −3.77001289397788834315564407000, −2.33053719732637116444902800732, −0.987084397963553541140627682006, 1.56653288968872296261032736297, 3.03887277067339483958184543626, 4.22405784595058421545279841596, 5.40483601606981130343617181781, 6.73152957487231431816802423343, 7.71086194561064245683798066980, 8.148082736829142478126543572821, 9.099246555549651469418294821309, 10.20468921731290505320204654483, 10.72882788588159851754110405464

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