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Properties

Label 2-540-12.11-c1-0-14
Degree $2$
Conductor $540$
Sign $0.251 + 0.967i$
Analytic cond. $4.31192$
Root an. cond. $2.07651$
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  + (−0.178 − 1.40i)2-s + (−1.93 + 0.502i)4-s + i·5-s − 1.60i·7-s + (1.05 + 2.62i)8-s + (1.40 − 0.178i)10-s + 6.10·11-s − 1.35·13-s + (−2.25 + 0.287i)14-s + (3.49 − 1.94i)16-s + 3.77i·17-s − 4.37i·19-s + (−0.502 − 1.93i)20-s + (−1.09 − 8.56i)22-s + 6.23·23-s + ⋯
L(s)  = 1  + (−0.126 − 0.991i)2-s + (−0.967 + 0.251i)4-s + 0.447i·5-s − 0.606i·7-s + (0.371 + 0.928i)8-s + (0.443 − 0.0565i)10-s + 1.84·11-s − 0.375·13-s + (−0.601 + 0.0767i)14-s + (0.873 − 0.486i)16-s + 0.916i·17-s − 1.00i·19-s + (−0.112 − 0.432i)20-s + (−0.232 − 1.82i)22-s + 1.29·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.251 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.251 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(540\)    =    \(2^{2} \cdot 3^{3} \cdot 5\)
Sign: $0.251 + 0.967i$
Analytic conductor: \(4.31192\)
Root analytic conductor: \(2.07651\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{540} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 540,\ (\ :1/2),\ 0.251 + 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04964 - 0.812116i\)
\(L(\frac12)\) \(\approx\) \(1.04964 - 0.812116i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.178 + 1.40i)T \)
3 \( 1 \)
5 \( 1 - iT \)
good7 \( 1 + 1.60iT - 7T^{2} \)
11 \( 1 - 6.10T + 11T^{2} \)
13 \( 1 + 1.35T + 13T^{2} \)
17 \( 1 - 3.77iT - 17T^{2} \)
19 \( 1 + 4.37iT - 19T^{2} \)
23 \( 1 - 6.23T + 23T^{2} \)
29 \( 1 + 7.15iT - 29T^{2} \)
31 \( 1 + 8.57iT - 31T^{2} \)
37 \( 1 - 6.37T + 37T^{2} \)
41 \( 1 - 2.07iT - 41T^{2} \)
43 \( 1 - 3.63iT - 43T^{2} \)
47 \( 1 + 0.484T + 47T^{2} \)
53 \( 1 - 3.20iT - 53T^{2} \)
59 \( 1 + 2.34T + 59T^{2} \)
61 \( 1 + 8.22T + 61T^{2} \)
67 \( 1 - 13.8iT - 67T^{2} \)
71 \( 1 - 12.5T + 71T^{2} \)
73 \( 1 - 2.20T + 73T^{2} \)
79 \( 1 + 9.52iT - 79T^{2} \)
83 \( 1 + 2.47T + 83T^{2} \)
89 \( 1 - 14.7iT - 89T^{2} \)
97 \( 1 + 18.0T + 97T^{2} \)
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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

−10.84846798299341089862102497540, −9.683719625204948120368343262941, −9.262953853425825608313409891966, −8.111668457331849117566194062344, −7.06690340116006067238442770099, −6.05603911616671237276897630990, −4.47602926848997199068931929352, −3.84868091023076912383441982430, −2.55587123359048084889855566456, −1.05956882492202005678596634706, 1.29307125265987411861034449276, 3.44580337887837014818275340533, 4.66011247408136311797461345370, 5.48566214516476711094917715534, 6.56391585095661071064187583586, 7.23781631057543908974836051062, 8.481776071568388834024766230178, 9.106798233250442192336033509414, 9.643550781175513498832235125739, 10.94925947078563245497824833478

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