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Properties

Label 2-165-11.9-c1-0-7
Degree $2$
Conductor $165$
Sign $-0.803 + 0.595i$
Analytic cond. $1.31753$
Root an. cond. $1.14783$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.788 − 2.42i)2-s + (0.809 − 0.587i)3-s + (−3.64 − 2.65i)4-s + (−0.309 − 0.951i)5-s + (−0.788 − 2.42i)6-s + (3.39 + 2.46i)7-s + (−5.18 + 3.76i)8-s + (0.309 − 0.951i)9-s − 2.55·10-s + (−2.04 + 2.61i)11-s − 4.51·12-s + (−1.52 + 4.69i)13-s + (8.65 − 6.29i)14-s + (−0.809 − 0.587i)15-s + (2.26 + 6.96i)16-s + (−1.88 − 5.80i)17-s + ⋯
L(s)  = 1  + (0.557 − 1.71i)2-s + (0.467 − 0.339i)3-s + (−1.82 − 1.32i)4-s + (−0.138 − 0.425i)5-s + (−0.321 − 0.990i)6-s + (1.28 + 0.931i)7-s + (−1.83 + 1.33i)8-s + (0.103 − 0.317i)9-s − 0.806·10-s + (−0.616 + 0.787i)11-s − 1.30·12-s + (−0.422 + 1.30i)13-s + (2.31 − 1.68i)14-s + (−0.208 − 0.151i)15-s + (0.565 + 1.74i)16-s + (−0.457 − 1.40i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.803 + 0.595i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.803 + 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $-0.803 + 0.595i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ -0.803 + 0.595i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.499171 - 1.51013i\)
\(L(\frac12)\) \(\approx\) \(0.499171 - 1.51013i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (2.04 - 2.61i)T \)
good2 \( 1 + (-0.788 + 2.42i)T + (-1.61 - 1.17i)T^{2} \)
7 \( 1 + (-3.39 - 2.46i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (1.52 - 4.69i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.88 + 5.80i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-2.03 + 1.47i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 3.08T + 23T^{2} \)
29 \( 1 + (-2.91 - 2.11i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.616 - 1.89i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (6.18 + 4.49i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-2.39 + 1.73i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 0.186T + 43T^{2} \)
47 \( 1 + (4.50 - 3.27i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.92 - 5.91i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-0.429 - 0.311i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.51 - 7.75i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 6.64T + 67T^{2} \)
71 \( 1 + (3.40 + 10.4i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.0361 + 0.0262i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-3.87 + 11.9i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.99 + 6.15i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 0.535T + 89T^{2} \)
97 \( 1 + (3.41 - 10.5i)T + (-78.4 - 57.0i)T^{2} \)
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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

−12.07244259784529709785396304441, −11.84513385498023484455655766247, −10.77939383010635121852323435625, −9.346854007837148131428955895541, −8.891587801767719791908014595620, −7.35990309699328280927373205723, −5.10387739178080037894594848026, −4.58467990070287301405440645008, −2.70667526512363952443758710372, −1.73827743026845114653568451604, 3.47861613114903420019409404977, 4.67744637216031838619273877255, 5.65449151931581738527535110903, 7.05576964656564620087456387322, 8.163743239273181869902736026163, 8.240509834372608813937181785826, 10.16902369293584081788918827829, 11.09201387481652213388350174796, 12.84292135504527536002211966827, 13.64737335129341222522204567573

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