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Properties

Label 2-546-91.24-c1-0-10
Degree $2$
Conductor $546$
Sign $0.947 + 0.318i$
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 + i·3-s + (0.866 − 0.499i)4-s + (−1.66 − 0.445i)5-s + (0.258 + 0.965i)6-s + (2.48 − 0.914i)7-s + (0.707 − 0.707i)8-s − 9-s − 1.71·10-s + (3.98 − 3.98i)11-s + (0.499 + 0.866i)12-s + (−1.62 − 3.22i)13-s + (2.16 − 1.52i)14-s + (0.445 − 1.66i)15-s + (0.500 − 0.866i)16-s + (3.82 + 6.63i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + 0.577i·3-s + (0.433 − 0.249i)4-s + (−0.742 − 0.199i)5-s + (0.105 + 0.394i)6-s + (0.938 − 0.345i)7-s + (0.249 − 0.249i)8-s − 0.333·9-s − 0.543·10-s + (1.20 − 1.20i)11-s + (0.144 + 0.249i)12-s + (−0.449 − 0.893i)13-s + (0.577 − 0.407i)14-s + (0.114 − 0.428i)15-s + (0.125 − 0.216i)16-s + (0.928 + 1.60i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.947 + 0.318i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.947 + 0.318i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.16379 - 0.354368i\)
\(L(\frac12)\) \(\approx\) \(2.16379 - 0.354368i\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
7 \( 1 + (-2.48 + 0.914i)T \)
13 \( 1 + (1.62 + 3.22i)T \)
good5 \( 1 + (1.66 + 0.445i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-3.98 + 3.98i)T - 11iT^{2} \)
17 \( 1 + (-3.82 - 6.63i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.256 - 0.256i)T - 19iT^{2} \)
23 \( 1 + (-5.49 - 3.17i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.90 - 6.76i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.0384 - 0.143i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (1.81 + 6.76i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (8.72 + 2.33i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (6.41 + 3.70i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.195 + 0.730i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.431 + 0.747i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.335 + 1.25i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 - 7.79iT - 61T^{2} \)
67 \( 1 + (2.36 + 2.36i)T + 67iT^{2} \)
71 \( 1 + (14.4 - 3.87i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (8.44 - 2.26i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-6.10 - 10.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.17 + 1.17i)T - 83iT^{2} \)
89 \( 1 + (-4.04 + 1.08i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-0.740 - 2.76i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.80350318598520062157052298554, −10.27772185429434717087105894853, −8.797571165321191813932942880756, −8.210035857836224346752420054281, −7.13718605716416648359143387601, −5.82792588263327163146119426263, −5.04451125975439490582699597536, −3.88250960917772113132382274510, −3.36884944428908343003712620151, −1.28411088458941751780993953151, 1.63680277591647205654535335037, 2.97371921180630166408634117046, 4.43650816011427222712819615984, 4.97302212169357854197825636433, 6.49847130427713764159540008142, 7.16407540200127221733785202249, 7.84799283902888459648787976701, 8.947929846434655407696198703095, 9.956414208297222148551201019985, 11.50248285573655421684191805279

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