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Properties

Label 2-165-5.4-c5-0-11
Degree $2$
Conductor $165$
Sign $-0.742 - 0.670i$
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  + 0.886i·2-s + 9i·3-s + 31.2·4-s + (−37.4 + 41.4i)5-s − 7.97·6-s − 224. i·7-s + 56.0i·8-s − 81·9-s + (−36.7 − 33.2i)10-s + 121·11-s + 280. i·12-s + 1.06e3i·13-s + 199.·14-s + (−373. − 337. i)15-s + 949.·16-s + 1.34e3i·17-s + ⋯
L(s)  = 1  + 0.156i·2-s + 0.577i·3-s + 0.975·4-s + (−0.670 + 0.742i)5-s − 0.0904·6-s − 1.73i·7-s + 0.309i·8-s − 0.333·9-s + (−0.116 − 0.105i)10-s + 0.301·11-s + 0.563i·12-s + 1.74i·13-s + 0.271·14-s + (−0.428 − 0.386i)15-s + 0.926·16-s + 1.13i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.742 - 0.670i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.742 - 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $-0.742 - 0.670i$
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: \((2,\ 165,\ (\ :5/2),\ -0.742 - 0.670i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.552054099\)
\(L(\frac12)\) \(\approx\) \(1.552054099\)
\(L(\frac{7}{2})\) 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 \( 1 + (37.4 - 41.4i)T \)
11 \( 1 - 121T \)
good2 \( 1 - 0.886iT - 32T^{2} \)
7 \( 1 + 224. iT - 1.68e4T^{2} \)
13 \( 1 - 1.06e3iT - 3.71e5T^{2} \)
17 \( 1 - 1.34e3iT - 1.41e6T^{2} \)
19 \( 1 + 691.T + 2.47e6T^{2} \)
23 \( 1 - 3.39e3iT - 6.43e6T^{2} \)
29 \( 1 + 8.60e3T + 2.05e7T^{2} \)
31 \( 1 + 320.T + 2.86e7T^{2} \)
37 \( 1 - 1.90e3iT - 6.93e7T^{2} \)
41 \( 1 - 5.82e3T + 1.15e8T^{2} \)
43 \( 1 - 1.42e3iT - 1.47e8T^{2} \)
47 \( 1 - 6.45e3iT - 2.29e8T^{2} \)
53 \( 1 - 9.10e3iT - 4.18e8T^{2} \)
59 \( 1 + 1.02e4T + 7.14e8T^{2} \)
61 \( 1 + 3.42e4T + 8.44e8T^{2} \)
67 \( 1 - 6.84e4iT - 1.35e9T^{2} \)
71 \( 1 - 1.88e3T + 1.80e9T^{2} \)
73 \( 1 + 8.77e4iT - 2.07e9T^{2} \)
79 \( 1 - 5.24e4T + 3.07e9T^{2} \)
83 \( 1 - 5.86e4iT - 3.93e9T^{2} \)
89 \( 1 - 1.13e5T + 5.58e9T^{2} \)
97 \( 1 - 1.15e5iT - 8.58e9T^{2} \)
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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

−11.92426358715966531506934024951, −11.04977726246409309094614430792, −10.67949057355534884889138845940, −9.458027282289487301389486306333, −7.80574977997009675732203458752, −7.10926471642541954557294199647, −6.22342366723814700811652059383, −4.19812062256624341621917560217, −3.57339924847435332966709399645, −1.68276100254805566276460941124, 0.47383612897063054757685230444, 2.10229025148749214034808462318, 3.16002803081369546649939147738, 5.22470609693605163157261644403, 6.08476547323833897487039737119, 7.47594803311630619153508714939, 8.330795057977399503812748938058, 9.314359431703216846719942511019, 10.88749648201985105956090022529, 11.77064730649219645950878801844

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