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Properties

Label 2-165-15.14-c2-0-11
Degree $2$
Conductor $165$
Sign $0.978 - 0.206i$
Analytic cond. $4.49592$
Root an. cond. $2.12035$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.58·2-s + (−1.33 − 2.68i)3-s + 8.88·4-s + (4.83 + 1.25i)5-s + (4.80 + 9.63i)6-s + 1.00i·7-s − 17.5·8-s + (−5.41 + 7.18i)9-s + (−17.3 − 4.51i)10-s − 3.31i·11-s + (−11.8 − 23.8i)12-s + 24.8i·13-s − 3.61i·14-s + (−3.10 − 14.6i)15-s + 27.4·16-s − 6.01·17-s + ⋯
L(s)  = 1  − 1.79·2-s + (−0.446 − 0.894i)3-s + 2.22·4-s + (0.967 + 0.251i)5-s + (0.801 + 1.60i)6-s + 0.143i·7-s − 2.19·8-s + (−0.601 + 0.798i)9-s + (−1.73 − 0.451i)10-s − 0.301i·11-s + (−0.991 − 1.98i)12-s + 1.91i·13-s − 0.258i·14-s + (−0.206 − 0.978i)15-s + 1.71·16-s − 0.353·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.206i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.978 - 0.206i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.978 - 0.206i$
Analytic conductor: \(4.49592\)
Root analytic conductor: \(2.12035\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1),\ 0.978 - 0.206i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.620663 + 0.0649146i\)
\(L(\frac12)\) \(\approx\) \(0.620663 + 0.0649146i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.33 + 2.68i)T \)
5 \( 1 + (-4.83 - 1.25i)T \)
11 \( 1 + 3.31iT \)
good2 \( 1 + 3.58T + 4T^{2} \)
7 \( 1 - 1.00iT - 49T^{2} \)
13 \( 1 - 24.8iT - 169T^{2} \)
17 \( 1 + 6.01T + 289T^{2} \)
19 \( 1 - 19.5T + 361T^{2} \)
23 \( 1 - 25.3T + 529T^{2} \)
29 \( 1 - 38.2iT - 841T^{2} \)
31 \( 1 + 14.8T + 961T^{2} \)
37 \( 1 + 54.3iT - 1.36e3T^{2} \)
41 \( 1 + 24.4iT - 1.68e3T^{2} \)
43 \( 1 - 8.84iT - 1.84e3T^{2} \)
47 \( 1 - 55.0T + 2.20e3T^{2} \)
53 \( 1 - 18.0T + 2.80e3T^{2} \)
59 \( 1 - 9.10iT - 3.48e3T^{2} \)
61 \( 1 + 40.3T + 3.72e3T^{2} \)
67 \( 1 - 50.6iT - 4.48e3T^{2} \)
71 \( 1 - 105. iT - 5.04e3T^{2} \)
73 \( 1 + 23.5iT - 5.32e3T^{2} \)
79 \( 1 - 46.8T + 6.24e3T^{2} \)
83 \( 1 - 6.73T + 6.88e3T^{2} \)
89 \( 1 - 7.28iT - 7.92e3T^{2} \)
97 \( 1 - 147. iT - 9.40e3T^{2} \)
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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

−12.25309827700719735936555738916, −11.26395585441486567655582803489, −10.64924246616052420398207137839, −9.255447788072859102369077430343, −8.865838562725503068118412003109, −7.23894840805297368421414741718, −6.82456547301543946404695223460, −5.59742444762724407188084095112, −2.41186702696103062500758945565, −1.30846387567797378229601606685, 0.830419360112018018503345343463, 2.85758799684564944484235557250, 5.19828225453658490816908570247, 6.27892332304575070392333410335, 7.65763013129525933944151120667, 8.808031071143366281113757107657, 9.647134510102329406083305370076, 10.27386186901344762413239600402, 10.95627629266974726605268419309, 12.13116007306030278876673443389

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