Properties

Label 2-546-273.242-c1-0-26
Degree $2$
Conductor $546$
Sign $0.272 + 0.962i$
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 + (−1.51 + 0.831i)3-s + (0.866 + 0.499i)4-s + (−1.98 − 0.533i)5-s + (−1.68 + 0.410i)6-s + (−2.22 − 1.42i)7-s + (0.707 + 0.707i)8-s + (1.61 − 2.52i)9-s + (−1.78 − 1.02i)10-s + (3.44 − 0.922i)11-s + (−1.73 − 0.0392i)12-s + (−3.26 + 1.53i)13-s + (−1.78 − 1.95i)14-s + (3.46 − 0.844i)15-s + (0.500 + 0.866i)16-s + (3.54 − 6.14i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (−0.877 + 0.480i)3-s + (0.433 + 0.249i)4-s + (−0.889 − 0.238i)5-s + (−0.686 + 0.167i)6-s + (−0.841 − 0.539i)7-s + (0.249 + 0.249i)8-s + (0.538 − 0.842i)9-s + (−0.564 − 0.325i)10-s + (1.03 − 0.278i)11-s + (−0.499 − 0.0113i)12-s + (−0.904 + 0.426i)13-s + (−0.476 − 0.522i)14-s + (0.894 − 0.218i)15-s + (0.125 + 0.216i)16-s + (0.860 − 1.48i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.272 + 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.272 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.272 + 0.962i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (515, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.272 + 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.744088 - 0.562776i\)
\(L(\frac12)\) \(\approx\) \(0.744088 - 0.562776i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.51 - 0.831i)T \)
7 \( 1 + (2.22 + 1.42i)T \)
13 \( 1 + (3.26 - 1.53i)T \)
good5 \( 1 + (1.98 + 0.533i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-3.44 + 0.922i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-3.54 + 6.14i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.49 + 5.57i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.12 + 5.42i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 0.340iT - 29T^{2} \)
31 \( 1 + (8.31 - 2.22i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-7.50 - 2.01i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-5.27 + 5.27i)T - 41iT^{2} \)
43 \( 1 + 0.428iT - 43T^{2} \)
47 \( 1 + (0.403 - 1.50i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (5.74 + 3.31i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.94 - 0.522i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (0.887 + 1.53i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (11.7 - 3.14i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (1.49 - 1.49i)T - 71iT^{2} \)
73 \( 1 + (1.14 + 4.28i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (4.48 + 7.76i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.13 - 8.13i)T - 83iT^{2} \)
89 \( 1 + (0.601 - 2.24i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-11.9 - 11.9i)T + 97iT^{2} \)
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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

−10.88860385547835207736821895338, −9.742516716150494249460353430888, −9.111479336420780974269867174957, −7.45184372755539259669544826597, −6.95012250959796935386015581263, −5.96318400884860640051675733596, −4.76502343297040230059951743970, −4.14739196721954249180687835061, −3.10035431699694835936578060434, −0.48742031305294817085992474448, 1.69521219467422943474885846724, 3.40326984338480420856403928767, 4.21251610524721799028661951747, 5.75718490436323611094908919875, 6.04416825701997186263089594045, 7.36094130699515420487247082645, 7.84783914796914100526503685460, 9.545309628907876401137705229868, 10.24685166893538268234246722303, 11.35530107081608618020758328216

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