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Properties

Label 2-165-5.4-c5-0-40
Degree $2$
Conductor $165$
Sign $0.662 + 0.749i$
Analytic cond. $26.4633$
Root an. cond. $5.14425$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8.30i·2-s − 9i·3-s − 37.0·4-s + (41.8 − 37.0i)5-s + 74.7·6-s − 54.6i·7-s − 41.5i·8-s − 81·9-s + (307. + 347. i)10-s − 121·11-s + 333. i·12-s + 175. i·13-s + 453.·14-s + (−333. − 377. i)15-s − 838.·16-s − 1.24e3i·17-s + ⋯
L(s)  = 1  + 1.46i·2-s − 0.577i·3-s − 1.15·4-s + (0.749 − 0.662i)5-s + 0.847·6-s − 0.421i·7-s − 0.229i·8-s − 0.333·9-s + (0.972 + 1.10i)10-s − 0.301·11-s + 0.667i·12-s + 0.287i·13-s + 0.618·14-s + (−0.382 − 0.432i)15-s − 0.819·16-s − 1.04i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.662 + 0.749i$
Analytic conductor: \(26.4633\)
Root analytic conductor: \(5.14425\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :5/2),\ 0.662 + 0.749i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.293258554\)
\(L(\frac12)\) \(\approx\) \(1.293258554\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 9iT \)
5 \( 1 + (-41.8 + 37.0i)T \)
11 \( 1 + 121T \)
good2 \( 1 - 8.30iT - 32T^{2} \)
7 \( 1 + 54.6iT - 1.68e4T^{2} \)
13 \( 1 - 175. iT - 3.71e5T^{2} \)
17 \( 1 + 1.24e3iT - 1.41e6T^{2} \)
19 \( 1 + 1.24e3T + 2.47e6T^{2} \)
23 \( 1 + 2.56e3iT - 6.43e6T^{2} \)
29 \( 1 + 3.45e3T + 2.05e7T^{2} \)
31 \( 1 - 3.17e3T + 2.86e7T^{2} \)
37 \( 1 + 1.53e4iT - 6.93e7T^{2} \)
41 \( 1 + 1.45e4T + 1.15e8T^{2} \)
43 \( 1 - 7.67e3iT - 1.47e8T^{2} \)
47 \( 1 + 2.17e4iT - 2.29e8T^{2} \)
53 \( 1 - 3.89e4iT - 4.18e8T^{2} \)
59 \( 1 - 4.60e3T + 7.14e8T^{2} \)
61 \( 1 + 5.62e3T + 8.44e8T^{2} \)
67 \( 1 + 3.55e4iT - 1.35e9T^{2} \)
71 \( 1 - 3.34e4T + 1.80e9T^{2} \)
73 \( 1 + 1.85e4iT - 2.07e9T^{2} \)
79 \( 1 + 1.56e4T + 3.07e9T^{2} \)
83 \( 1 + 7.62e4iT - 3.93e9T^{2} \)
89 \( 1 - 5.92e4T + 5.58e9T^{2} \)
97 \( 1 + 7.72e4iT - 8.58e9T^{2} \)
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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

−12.08990664793840315178752137569, −10.66745945536231660201354556046, −9.284219565656069779005306803545, −8.484766128515079426620710417223, −7.42114958433946171874454538275, −6.54107028558820134112275664108, −5.57443643315081126906287099213, −4.54388817137271494963398598494, −2.20164288416821684310813814549, −0.39865681790322087696875856251, 1.65749062803393958984217743147, 2.76446443576999849025776621787, 3.80552586310523968799637153231, 5.30752398099801644472225638312, 6.56523045983752817854088419087, 8.380311640798505744795606594974, 9.556162791286155679816329588458, 10.22068136506681986038704818646, 10.92822442487414870814080106121, 11.81595272937055221314424774524

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