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Properties

Label 2-165-5.4-c3-0-8
Degree $2$
Conductor $165$
Sign $-0.703 + 0.710i$
Analytic cond. $9.73531$
Root an. cond. $3.12014$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.31i·2-s + 3i·3-s − 10.5·4-s + (7.94 + 7.86i)5-s − 12.9·6-s + 9.06i·7-s − 11.1i·8-s − 9·9-s + (−33.9 + 34.2i)10-s − 11·11-s − 31.7i·12-s + 40.4i·13-s − 39.0·14-s + (−23.5 + 23.8i)15-s − 36.5·16-s − 76.4i·17-s + ⋯
L(s)  = 1  + 1.52i·2-s + 0.577i·3-s − 1.32·4-s + (0.710 + 0.703i)5-s − 0.880·6-s + 0.489i·7-s − 0.494i·8-s − 0.333·9-s + (−1.07 + 1.08i)10-s − 0.301·11-s − 0.764i·12-s + 0.862i·13-s − 0.746·14-s + (−0.406 + 0.410i)15-s − 0.570·16-s − 1.09i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.703 + 0.710i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.703 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $-0.703 + 0.710i$
Analytic conductor: \(9.73531\)
Root analytic conductor: \(3.12014\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :3/2),\ -0.703 + 0.710i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.601394 - 1.44169i\)
\(L(\frac12)\) \(\approx\) \(0.601394 - 1.44169i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3iT \)
5 \( 1 + (-7.94 - 7.86i)T \)
11 \( 1 + 11T \)
good2 \( 1 - 4.31iT - 8T^{2} \)
7 \( 1 - 9.06iT - 343T^{2} \)
13 \( 1 - 40.4iT - 2.19e3T^{2} \)
17 \( 1 + 76.4iT - 4.91e3T^{2} \)
19 \( 1 - 44.7T + 6.85e3T^{2} \)
23 \( 1 + 50.4iT - 1.21e4T^{2} \)
29 \( 1 - 81.1T + 2.43e4T^{2} \)
31 \( 1 - 100.T + 2.97e4T^{2} \)
37 \( 1 + 38.7iT - 5.06e4T^{2} \)
41 \( 1 - 18.6T + 6.89e4T^{2} \)
43 \( 1 - 40.8iT - 7.95e4T^{2} \)
47 \( 1 + 36.4iT - 1.03e5T^{2} \)
53 \( 1 - 652. iT - 1.48e5T^{2} \)
59 \( 1 + 744.T + 2.05e5T^{2} \)
61 \( 1 + 516.T + 2.26e5T^{2} \)
67 \( 1 - 430. iT - 3.00e5T^{2} \)
71 \( 1 + 272.T + 3.57e5T^{2} \)
73 \( 1 - 692. iT - 3.89e5T^{2} \)
79 \( 1 - 1.08e3T + 4.93e5T^{2} \)
83 \( 1 - 595. iT - 5.71e5T^{2} \)
89 \( 1 - 706.T + 7.04e5T^{2} \)
97 \( 1 - 1.31e3iT - 9.12e5T^{2} \)
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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

−13.68205903651479250352464797375, −11.96863474825035415395670739154, −10.82322179866411445842296830486, −9.586541283934289512093074813187, −8.903199985865390190048742257788, −7.57064414121072052199907557251, −6.58017915753225764125825811884, −5.67018123310457530842279716456, −4.64505224909234966873854553151, −2.67261792849289858284293578949, 0.74246766381527953038832249684, 1.87642891791202600340846309712, 3.29461736310518070961594161578, 4.81502445358006632943226726006, 6.17681537588991297010583667968, 7.81410990197283398945333310618, 8.946922344870155852282353735202, 10.05378806976034756140632670606, 10.67681555910333400075562588505, 11.90117765365543699215208806675

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