Properties

Label 2-165-165.32-c0-0-0
Degree $2$
Conductor $165$
Sign $0.850 - 0.525i$
Analytic cond. $0.0823457$
Root an. cond. $0.286959$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s i·4-s + i·5-s − 9-s i·11-s + 12-s − 15-s − 16-s + 20-s + (−1 − i)23-s − 25-s i·27-s + 33-s + i·36-s + (1 + i)37-s + ⋯
L(s)  = 1  + i·3-s i·4-s + i·5-s − 9-s i·11-s + 12-s − 15-s − 16-s + 20-s + (−1 − i)23-s − 25-s i·27-s + 33-s + i·36-s + (1 + i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.850 - 0.525i$
Analytic conductor: \(0.0823457\)
Root analytic conductor: \(0.286959\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :0),\ 0.850 - 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6396858569\)
\(L(\frac12)\) \(\approx\) \(0.6396858569\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
5 \( 1 - iT \)
11 \( 1 + iT \)
good2 \( 1 + iT^{2} \)
7 \( 1 - iT^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (1 + i)T + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-1 - i)T + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (1 - i)T - iT^{2} \)
53 \( 1 + (-1 - i)T + iT^{2} \)
59 \( 1 - 2T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (1 + i)T + iT^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (1 + i)T + iT^{2} \)
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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

−13.58387184974898777730455684846, −11.71565787232289332281999171781, −10.91510297876007671294013685694, −10.28540261466507186940224189090, −9.416118499399610651969579898083, −8.187516102217810345877015148760, −6.46389700689867982798864646199, −5.70988805836185351027648661991, −4.27941801431036193310475328141, −2.78769295406376910388561306770, 2.09761890952711296516119355548, 3.95015737579757376172805773740, 5.42246871145185096342319339715, 6.93281965314840913390216512087, 7.83096366481836823171257604871, 8.616384978317592918256578488262, 9.728772459251592395121557522403, 11.57875433332852445614781140229, 12.08663318021246051986637663717, 12.99948646994780214470491825403

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