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Properties

Label 2-165-11.5-c1-0-2
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: $\chi_{165} (16, \cdot )$
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

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

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