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Properties

Label 2-135-5.3-c2-0-2
Degree $2$
Conductor $135$
Sign $-0.955 - 0.293i$
Analytic cond. $3.67848$
Root an. cond. $1.91793$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.683 + 0.683i)2-s + 3.06i·4-s + (−2.52 + 4.31i)5-s + (1.91 − 1.91i)7-s + (−4.82 − 4.82i)8-s + (−1.22 − 4.67i)10-s − 16.0·11-s + (0.607 + 0.607i)13-s + 2.61i·14-s − 5.66·16-s + (−7.56 + 7.56i)17-s + 21.0i·19-s + (−13.2 − 7.75i)20-s + (10.9 − 10.9i)22-s + (25.2 + 25.2i)23-s + ⋯
L(s)  = 1  + (−0.341 + 0.341i)2-s + 0.766i·4-s + (−0.505 + 0.862i)5-s + (0.273 − 0.273i)7-s + (−0.603 − 0.603i)8-s + (−0.122 − 0.467i)10-s − 1.45·11-s + (0.0467 + 0.0467i)13-s + 0.186i·14-s − 0.354·16-s + (−0.444 + 0.444i)17-s + 1.10i·19-s + (−0.661 − 0.387i)20-s + (0.498 − 0.498i)22-s + (1.09 + 1.09i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 - 0.293i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.955 - 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $-0.955 - 0.293i$
Analytic conductor: \(3.67848\)
Root analytic conductor: \(1.91793\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{135} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 135,\ (\ :1),\ -0.955 - 0.293i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.103070 + 0.686067i\)
\(L(\frac12)\) \(\approx\) \(0.103070 + 0.686067i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (2.52 - 4.31i)T \)
good2 \( 1 + (0.683 - 0.683i)T - 4iT^{2} \)
7 \( 1 + (-1.91 + 1.91i)T - 49iT^{2} \)
11 \( 1 + 16.0T + 121T^{2} \)
13 \( 1 + (-0.607 - 0.607i)T + 169iT^{2} \)
17 \( 1 + (7.56 - 7.56i)T - 289iT^{2} \)
19 \( 1 - 21.0iT - 361T^{2} \)
23 \( 1 + (-25.2 - 25.2i)T + 529iT^{2} \)
29 \( 1 + 13.4iT - 841T^{2} \)
31 \( 1 + 39.2T + 961T^{2} \)
37 \( 1 + (-5.47 + 5.47i)T - 1.36e3iT^{2} \)
41 \( 1 - 69.4T + 1.68e3T^{2} \)
43 \( 1 + (-55.6 - 55.6i)T + 1.84e3iT^{2} \)
47 \( 1 + (-13.7 + 13.7i)T - 2.20e3iT^{2} \)
53 \( 1 + (-34.8 - 34.8i)T + 2.80e3iT^{2} \)
59 \( 1 - 62.0iT - 3.48e3T^{2} \)
61 \( 1 + 67.2T + 3.72e3T^{2} \)
67 \( 1 + (-29.2 + 29.2i)T - 4.48e3iT^{2} \)
71 \( 1 + 71.3T + 5.04e3T^{2} \)
73 \( 1 + (23.5 + 23.5i)T + 5.32e3iT^{2} \)
79 \( 1 - 5.61iT - 6.24e3T^{2} \)
83 \( 1 + (26.0 + 26.0i)T + 6.88e3iT^{2} \)
89 \( 1 - 23.2iT - 7.92e3T^{2} \)
97 \( 1 + (-97.3 + 97.3i)T - 9.40e3iT^{2} \)
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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

−13.33821339183932304242268971746, −12.50127002395672650705220684696, −11.26721219517520895200166619090, −10.52151655700568938758429409641, −9.119051214251778426451550698543, −7.67052406439138037205321831339, −7.56092577740206340417167054825, −5.95737219518420315485119736590, −4.10553126193122146566399299255, −2.81805364705634453381426278192, 0.50348031144543709441697563165, 2.47549666159035279967223394215, 4.73902359518812485646393706717, 5.51473225931288940947306669750, 7.26492183477160666783263532014, 8.607182249108369335392023360963, 9.255345784444892228366153344449, 10.66776065294930472710272343138, 11.24603558339712105509317773805, 12.53587171895048411327707494279

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