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Properties

Label 2-165-11.9-c1-0-5
Degree $2$
Conductor $165$
Sign $-0.151 + 0.988i$
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  + (0.564 − 1.73i)2-s + (0.809 − 0.587i)3-s + (−1.08 − 0.786i)4-s + (0.309 + 0.951i)5-s + (−0.564 − 1.73i)6-s + (−1.41 − 1.02i)7-s + (0.978 − 0.710i)8-s + (0.309 − 0.951i)9-s + 1.82·10-s + (−0.384 − 3.29i)11-s − 1.33·12-s + (−1.65 + 5.09i)13-s + (−2.58 + 1.87i)14-s + (0.809 + 0.587i)15-s + (−1.50 − 4.64i)16-s + (2.26 + 6.97i)17-s + ⋯
L(s)  = 1  + (0.399 − 1.22i)2-s + (0.467 − 0.339i)3-s + (−0.541 − 0.393i)4-s + (0.138 + 0.425i)5-s + (−0.230 − 0.709i)6-s + (−0.534 − 0.388i)7-s + (0.345 − 0.251i)8-s + (0.103 − 0.317i)9-s + 0.577·10-s + (−0.115 − 0.993i)11-s − 0.386·12-s + (−0.459 + 1.41i)13-s + (−0.690 + 0.501i)14-s + (0.208 + 0.151i)15-s + (−0.377 − 1.16i)16-s + (0.549 + 1.69i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.151 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.151 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $-0.151 + 0.988i$
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: \((2,\ 165,\ (\ :1/2),\ -0.151 + 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03020 - 1.20035i\)
\(L(\frac12)\) \(\approx\) \(1.03020 - 1.20035i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.384 + 3.29i)T \)
good2 \( 1 + (-0.564 + 1.73i)T + (-1.61 - 1.17i)T^{2} \)
7 \( 1 + (1.41 + 1.02i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (1.65 - 5.09i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-2.26 - 6.97i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (4.86 - 3.53i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 8.35T + 23T^{2} \)
29 \( 1 + (4.48 + 3.25i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.00660 - 0.0203i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-0.907 - 0.659i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-2.96 + 2.15i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 1.96T + 43T^{2} \)
47 \( 1 + (3.09 - 2.24i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-0.516 + 1.58i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (4.58 + 3.32i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.86 - 8.82i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 0.350T + 67T^{2} \)
71 \( 1 + (-1.40 - 4.31i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (8.20 + 5.95i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-0.792 + 2.43i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (4.96 + 15.2i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 14.5T + 89T^{2} \)
97 \( 1 + (0.587 - 1.80i)T + (-78.4 - 57.0i)T^{2} \)
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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

−12.71073920313072027342913114418, −11.54742796190268683729361778269, −10.71757327713142724514388431660, −9.851272795914726404350517242082, −8.652616660176047033046473990681, −7.25783039518800600877332376153, −6.16601506652820738187298512876, −4.14376349082546832384167655365, −3.21837081406294947857331188249, −1.80265091836704642962614085711, 2.78793146791899001230763737823, 4.75047382837837870499722461840, 5.38911894988061274528227024586, 6.88639662899457039645261162277, 7.67742711746643367800777288145, 8.902686498960953329805601890451, 9.779489877001253528204297955673, 11.01698097526209754155040494365, 12.68084818730778192276344115509, 13.13386570539898242708061141318

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