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Properties

Label 2-546-273.257-c1-0-11
Degree $2$
Conductor $546$
Sign $0.910 - 0.412i$
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 + (1.66 − 0.492i)3-s + 4-s + (1.80 + 1.04i)5-s + (−1.66 + 0.492i)6-s + (−0.800 + 2.52i)7-s − 8-s + (2.51 − 1.63i)9-s + (−1.80 − 1.04i)10-s + (1.07 − 1.86i)11-s + (1.66 − 0.492i)12-s + (0.217 + 3.59i)13-s + (0.800 − 2.52i)14-s + (3.50 + 0.839i)15-s + 16-s − 0.557·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.958 − 0.284i)3-s + 0.5·4-s + (0.806 + 0.465i)5-s + (−0.677 + 0.201i)6-s + (−0.302 + 0.953i)7-s − 0.353·8-s + (0.838 − 0.545i)9-s + (−0.570 − 0.329i)10-s + (0.325 − 0.563i)11-s + (0.479 − 0.142i)12-s + (0.0602 + 0.998i)13-s + (0.213 − 0.673i)14-s + (0.905 + 0.216i)15-s + 0.250·16-s − 0.135·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.910 - 0.412i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.910 - 0.412i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61737 + 0.349310i\)
\(L(\frac12)\) \(\approx\) \(1.61737 + 0.349310i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.66 + 0.492i)T \)
7 \( 1 + (0.800 - 2.52i)T \)
13 \( 1 + (-0.217 - 3.59i)T \)
good5 \( 1 + (-1.80 - 1.04i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.07 + 1.86i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 0.557T + 17T^{2} \)
19 \( 1 + (-1.94 - 3.37i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 2.07iT - 23T^{2} \)
29 \( 1 + (6.10 - 3.52i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.21 - 5.57i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 8.31iT - 37T^{2} \)
41 \( 1 + (0.532 - 0.307i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.33 + 7.50i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.507 + 0.292i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.68 + 3.85i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 1.56iT - 59T^{2} \)
61 \( 1 + (7.10 - 4.10i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-12.3 - 7.12i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.52 - 11.3i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.198 - 0.344i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.73 + 9.93i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 12.4iT - 83T^{2} \)
89 \( 1 + 8.70iT - 89T^{2} \)
97 \( 1 + (-1.64 + 2.85i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.57395268529227303131831203085, −9.729787371128267003325091549939, −9.016252164949866405370246158498, −8.542811991901158786245348297482, −7.29422398389410091950465051154, −6.50516685126834858532108523390, −5.64664683185002825563215018613, −3.75908108901051762859949468722, −2.60154074181411291198184108226, −1.72847064197837172231619726154, 1.25436041169096656686327910004, 2.60535815398715891422617382045, 3.84124949914012967641198420177, 5.05942592056548311979833869873, 6.38799545445637185945230381833, 7.48802644802021479647214463808, 8.046504741147019201282426733430, 9.377478020855768359483160717589, 9.548359869846615480563252600223, 10.37253285472658954182599442490

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