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Properties

Label 2-135-9.5-c2-0-2
Degree $2$
Conductor $135$
Sign $0.641 - 0.767i$
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  + (−2.67 + 1.54i)2-s + (2.77 − 4.81i)4-s + (−1.93 − 1.11i)5-s + (1.10 + 1.91i)7-s + 4.80i·8-s + 6.91·10-s + (15.4 − 8.91i)11-s + (1.25 − 2.17i)13-s + (−5.90 − 3.40i)14-s + (3.67 + 6.37i)16-s + 32.6i·17-s + 7.93·19-s + (−10.7 + 6.21i)20-s + (−27.5 + 47.7i)22-s + (18.4 + 10.6i)23-s + ⋯
L(s)  = 1  + (−1.33 + 0.772i)2-s + (0.694 − 1.20i)4-s + (−0.387 − 0.223i)5-s + (0.157 + 0.272i)7-s + 0.601i·8-s + 0.691·10-s + (1.40 − 0.810i)11-s + (0.0966 − 0.167i)13-s + (−0.421 − 0.243i)14-s + (0.229 + 0.398i)16-s + 1.91i·17-s + 0.417·19-s + (−0.537 + 0.310i)20-s + (−1.25 + 2.17i)22-s + (0.801 + 0.462i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.641 - 0.767i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.641 - 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $0.641 - 0.767i$
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: \((2,\ 135,\ (\ :1),\ 0.641 - 0.767i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.667577 + 0.312212i\)
\(L(\frac12)\) \(\approx\) \(0.667577 + 0.312212i\)
\(L(2)\) 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 \( 1 + (2.67 - 1.54i)T + (2 - 3.46i)T^{2} \)
7 \( 1 + (-1.10 - 1.91i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-15.4 + 8.91i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-1.25 + 2.17i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 32.6iT - 289T^{2} \)
19 \( 1 - 7.93T + 361T^{2} \)
23 \( 1 + (-18.4 - 10.6i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-30.7 + 17.7i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (1.01 - 1.75i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 50.6T + 1.36e3T^{2} \)
41 \( 1 + (4.65 + 2.68i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (7.76 + 13.4i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-1.63 + 0.943i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 62.0iT - 2.80e3T^{2} \)
59 \( 1 + (31.3 + 18.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-21.1 - 36.5i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-7.38 + 12.7i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 105. iT - 5.04e3T^{2} \)
73 \( 1 + 66.9T + 5.32e3T^{2} \)
79 \( 1 + (-34.5 - 59.8i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-18.3 + 10.6i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 7.16iT - 7.92e3T^{2} \)
97 \( 1 + (55.6 + 96.4i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−13.13393050432509137149133456644, −11.86996411513131577095339769601, −10.87029682953633922900201312607, −9.698594728417999993347044496745, −8.674821462870110040203771338924, −8.149442850449227355276556709364, −6.79223263605157096053545588210, −5.85736325773102479074492312927, −3.84733502381457299361656610155, −1.15992521836346367086397223227, 1.06649051308553459756556039397, 2.86781800405798288038098654520, 4.61095301618316541354628781693, 6.82231098836579554973960320206, 7.66547060457785130621371210550, 9.038508944480640100516859677214, 9.582708361000478912467076262798, 10.77299038870374082170521543502, 11.64001101441626395453714325693, 12.24308884250389095265470887743

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