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Properties

Label 2-165-5.4-c1-0-0
Degree $2$
Conductor $165$
Sign $-0.749 + 0.662i$
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  + 2.67i·2-s + i·3-s − 5.15·4-s + (−1.48 − 1.67i)5-s − 2.67·6-s + 2.80i·7-s − 8.44i·8-s − 9-s + (4.48 − 3.96i)10-s − 11-s − 5.15i·12-s + 5.11i·13-s − 7.50·14-s + (1.67 − 1.48i)15-s + 12.2·16-s + 4.54i·17-s + ⋯
L(s)  = 1  + 1.89i·2-s + 0.577i·3-s − 2.57·4-s + (−0.662 − 0.749i)5-s − 1.09·6-s + 1.06i·7-s − 2.98i·8-s − 0.333·9-s + (1.41 − 1.25i)10-s − 0.301·11-s − 1.48i·12-s + 1.41i·13-s − 2.00·14-s + (0.432 − 0.382i)15-s + 3.06·16-s + 1.10i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $-0.749 + 0.662i$
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: \((2,\ 165,\ (\ :1/2),\ -0.749 + 0.662i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.261043 - 0.689302i\)
\(L(\frac12)\) \(\approx\) \(0.261043 - 0.689302i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.48 + 1.67i)T \)
11 \( 1 + T \)
good2 \( 1 - 2.67iT - 2T^{2} \)
7 \( 1 - 2.80iT - 7T^{2} \)
13 \( 1 - 5.11iT - 13T^{2} \)
17 \( 1 - 4.54iT - 17T^{2} \)
19 \( 1 - 4.57T + 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 - 2.38T + 29T^{2} \)
31 \( 1 + 0.962T + 31T^{2} \)
37 \( 1 - 1.61iT - 37T^{2} \)
41 \( 1 + 2.38T + 41T^{2} \)
43 \( 1 + 2.80iT - 43T^{2} \)
47 \( 1 + 4.31iT - 47T^{2} \)
53 \( 1 - 6.57iT - 53T^{2} \)
59 \( 1 - 13.2T + 59T^{2} \)
61 \( 1 - 7.92T + 61T^{2} \)
67 \( 1 - 10.7iT - 67T^{2} \)
71 \( 1 + 7.35T + 71T^{2} \)
73 \( 1 + 6.41iT - 73T^{2} \)
79 \( 1 + 1.35T + 79T^{2} \)
83 \( 1 - 0.806iT - 83T^{2} \)
89 \( 1 - 2.96T + 89T^{2} \)
97 \( 1 + 9.92iT - 97T^{2} \)
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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

−13.80815981188055901181247167652, −12.67463071432201129120086936914, −11.70023716018659371518966697725, −9.872878449640682432181617892238, −8.775269894403588681686992722522, −8.500038199531680390410089957856, −7.16612056023391829134103647772, −5.91386979257799524254226600643, −4.99000437278833084555213977802, −4.01008473251457653936656619490, 0.74384208295719813699571457906, 2.80232228106684784750289440788, 3.66456603395591781409371144511, 5.18625527457841048396557464908, 7.29384409681902946094661911268, 8.132716541248708397467615746452, 9.697126543035347835075532823636, 10.48332802809698717519303092169, 11.28340463605934894748849844942, 11.98193134693231891890697633425

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