Properties

Label 2-165-5.4-c1-0-9
Degree $2$
Conductor $165$
Sign $0.990 + 0.139i$
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  + 1.21i·2-s i·3-s + 0.525·4-s + (0.311 − 2.21i)5-s + 1.21·6-s − 4.90i·7-s + 3.06i·8-s − 9-s + (2.68 + 0.377i)10-s − 11-s − 0.525i·12-s + 4.14i·13-s + 5.95·14-s + (−2.21 − 0.311i)15-s − 2.67·16-s + 5.33i·17-s + ⋯
L(s)  = 1  + 0.858i·2-s − 0.577i·3-s + 0.262·4-s + (0.139 − 0.990i)5-s + 0.495·6-s − 1.85i·7-s + 1.08i·8-s − 0.333·9-s + (0.850 + 0.119i)10-s − 0.301·11-s − 0.151i·12-s + 1.15i·13-s + 1.59·14-s + (−0.571 − 0.0803i)15-s − 0.668·16-s + 1.29i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.139i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.990 + 0.139i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ 0.990 + 0.139i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27678 - 0.0892547i\)
\(L(\frac12)\) \(\approx\) \(1.27678 - 0.0892547i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 + (-0.311 + 2.21i)T \)
11 \( 1 + T \)
good2 \( 1 - 1.21iT - 2T^{2} \)
7 \( 1 + 4.90iT - 7T^{2} \)
13 \( 1 - 4.14iT - 13T^{2} \)
17 \( 1 - 5.33iT - 17T^{2} \)
19 \( 1 - 5.18T + 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 + 1.80T + 29T^{2} \)
31 \( 1 - 2.62T + 31T^{2} \)
37 \( 1 + 5.80iT - 37T^{2} \)
41 \( 1 - 1.80T + 41T^{2} \)
43 \( 1 - 4.90iT - 43T^{2} \)
47 \( 1 + 7.05iT - 47T^{2} \)
53 \( 1 + 7.18iT - 53T^{2} \)
59 \( 1 + 1.67T + 59T^{2} \)
61 \( 1 - 0.755T + 61T^{2} \)
67 \( 1 - 4.85iT - 67T^{2} \)
71 \( 1 - 0.428T + 71T^{2} \)
73 \( 1 - 12.7iT - 73T^{2} \)
79 \( 1 - 6.42T + 79T^{2} \)
83 \( 1 + 2.90iT - 83T^{2} \)
89 \( 1 + 0.622T + 89T^{2} \)
97 \( 1 - 2.75iT - 97T^{2} \)
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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

−13.12120820285906345804783116867, −11.85165422693635068718812965093, −10.92134474258131848440946901879, −9.647003665823371858094714042611, −8.263296230260132690868536655374, −7.49660902549236248131626906629, −6.66430025550253030416712953255, −5.41566941104883907629370461008, −4.02601372739391660383617864153, −1.52248668343682865175322114389, 2.60156475798082844688044219292, 3.08580578737002228461828351961, 5.22825250951456454655985277584, 6.27550311638454957668389070865, 7.72235765685648655468329526974, 9.213291890312310662632329746362, 9.987531896765628996007532085677, 10.93673614159179793001807129298, 11.75274499451468769814497832711, 12.42591700326772259135131961979

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