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Properties

Label 2-540-180.119-c1-0-22
Degree $2$
Conductor $540$
Sign $0.874 + 0.484i$
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  + (1.25 − 0.645i)2-s + (1.16 − 1.62i)4-s + (1.98 + 1.03i)5-s + (−1.60 + 2.78i)7-s + (0.419 − 2.79i)8-s + (3.16 + 0.0199i)10-s + (1.56 − 2.70i)11-s + (1.59 − 0.923i)13-s + (−0.225 + 4.54i)14-s + (−1.27 − 3.79i)16-s + 5.85·17-s + 2.24i·19-s + (3.99 − 2.01i)20-s + (0.219 − 4.41i)22-s + (−3.24 + 1.87i)23-s + ⋯
L(s)  = 1  + (0.889 − 0.456i)2-s + (0.583 − 0.812i)4-s + (0.886 + 0.462i)5-s + (−0.608 + 1.05i)7-s + (0.148 − 0.988i)8-s + (0.999 + 0.00632i)10-s + (0.471 − 0.816i)11-s + (0.443 − 0.256i)13-s + (−0.0602 + 1.21i)14-s + (−0.319 − 0.947i)16-s + 1.41·17-s + 0.514i·19-s + (0.892 − 0.450i)20-s + (0.0467 − 0.941i)22-s + (−0.676 + 0.390i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 + 0.484i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.874 + 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(540\)    =    \(2^{2} \cdot 3^{3} \cdot 5\)
Sign: $0.874 + 0.484i$
Analytic conductor: \(4.31192\)
Root analytic conductor: \(2.07651\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{540} (359, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 540,\ (\ :1/2),\ 0.874 + 0.484i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.63413 - 0.681372i\)
\(L(\frac12)\) \(\approx\) \(2.63413 - 0.681372i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.25 + 0.645i)T \)
3 \( 1 \)
5 \( 1 + (-1.98 - 1.03i)T \)
good7 \( 1 + (1.60 - 2.78i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.56 + 2.70i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.59 + 0.923i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 5.85T + 17T^{2} \)
19 \( 1 - 2.24iT - 19T^{2} \)
23 \( 1 + (3.24 - 1.87i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (6.90 + 3.98i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.03 - 0.599i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.66iT - 37T^{2} \)
41 \( 1 + (-0.208 + 0.120i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.66 - 4.61i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.96 + 2.28i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 13.0T + 53T^{2} \)
59 \( 1 + (3.47 + 6.01i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.66 + 2.88i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.31 - 7.48i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.3T + 71T^{2} \)
73 \( 1 - 13.1iT - 73T^{2} \)
79 \( 1 + (11.6 + 6.72i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.07 + 0.623i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 14.3iT - 89T^{2} \)
97 \( 1 + (5.75 + 3.32i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.92628255338090478406948506502, −9.800159392742322135136577648619, −9.467010772422394257074962666799, −8.060897676548973352737539578309, −6.64521917472401264585262048411, −5.79276882055215448732345065344, −5.54334457064798058541849929458, −3.68911750567376040485185548778, −2.94339058538267833647362860640, −1.66136789040784350639989461629, 1.67758328098285985611129798449, 3.30662401188240522462434330294, 4.28855275156540058251770745429, 5.30285936379109089377239816660, 6.29839645206646241624861869111, 7.02203642751831766895331058718, 7.956836030250620371627982838162, 9.205399312857119673198379491519, 9.989346290128307913293572603542, 10.90653616973232184372804885510

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