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Properties

Label 2-135-9.5-c2-0-0
Degree $2$
Conductor $135$
Sign $-0.558 - 0.829i$
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  + (0.0412 − 0.0238i)2-s + (−1.99 + 3.46i)4-s + (−1.93 − 1.11i)5-s + (1.90 + 3.30i)7-s + 0.380i·8-s − 0.106·10-s + (−15.9 + 9.23i)11-s + (−8.96 + 15.5i)13-s + (0.157 + 0.0908i)14-s + (−7.98 − 13.8i)16-s + 9.22i·17-s + 26.4·19-s + (7.74 − 4.46i)20-s + (−0.439 + 0.761i)22-s + (−6.69 − 3.86i)23-s + ⋯
L(s)  = 1  + (0.0206 − 0.0119i)2-s + (−0.499 + 0.865i)4-s + (−0.387 − 0.223i)5-s + (0.272 + 0.471i)7-s + 0.0475i·8-s − 0.0106·10-s + (−1.45 + 0.839i)11-s + (−0.689 + 1.19i)13-s + (0.0112 + 0.00648i)14-s + (−0.499 − 0.864i)16-s + 0.542i·17-s + 1.39·19-s + (0.387 − 0.223i)20-s + (−0.0199 + 0.0346i)22-s + (−0.291 − 0.168i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.558 - 0.829i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.558 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $-0.558 - 0.829i$
Analytic conductor: \(3.67848\)
Root analytic conductor: \(1.91793\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{135} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 135,\ (\ :1),\ -0.558 - 0.829i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.390472 + 0.733483i\)
\(L(\frac12)\) \(\approx\) \(0.390472 + 0.733483i\)
\(L(2)\) 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 \( 1 + (1.93 + 1.11i)T \)
good2 \( 1 + (-0.0412 + 0.0238i)T + (2 - 3.46i)T^{2} \)
7 \( 1 + (-1.90 - 3.30i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (15.9 - 9.23i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (8.96 - 15.5i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 9.22iT - 289T^{2} \)
19 \( 1 - 26.4T + 361T^{2} \)
23 \( 1 + (6.69 + 3.86i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-20.1 + 11.6i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (0.881 - 1.52i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 10.0T + 1.36e3T^{2} \)
41 \( 1 + (-26.7 - 15.4i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (23.2 + 40.3i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-7.03 + 4.06i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 88.2iT - 2.80e3T^{2} \)
59 \( 1 + (-45.6 - 26.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-3.99 - 6.92i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (22.5 - 39.1i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 30.9iT - 5.04e3T^{2} \)
73 \( 1 + 51.5T + 5.32e3T^{2} \)
79 \( 1 + (-41.1 - 71.2i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (124. - 71.8i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 76.3iT - 7.92e3T^{2} \)
97 \( 1 + (-76.8 - 133. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−13.20735470801390153510054437435, −12.24998770864605766974424338954, −11.69731264873166802833391304103, −10.12985086112885039322515996561, −9.063660188716713636212140110812, −7.990624450492243294120690699017, −7.21830906975876346851126966176, −5.24135778900942422218825045734, −4.25936613433212388000997170805, −2.55557598898305893391542696083, 0.55538350742720061314939158411, 3.04553187695466065760587941459, 4.86929917962100446828610893764, 5.68695120749214426773667452891, 7.39499000972212899739036769461, 8.275695802029714549144133391502, 9.779616411192105395831467833175, 10.50260780266562969109935336996, 11.40697396050587577690535492993, 12.85771204144088692484343546510

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