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Properties

Label 2-165-5.4-c5-0-29
Degree $2$
Conductor $165$
Sign $0.521 - 0.853i$
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  + 4.97i·2-s − 9i·3-s + 7.25·4-s + (47.6 + 29.1i)5-s + 44.7·6-s − 155. i·7-s + 195. i·8-s − 81·9-s + (−145. + 237. i)10-s + 121·11-s − 65.3i·12-s + 1.17e3i·13-s + 773.·14-s + (262. − 429. i)15-s − 738.·16-s − 898. i·17-s + ⋯
L(s)  = 1  + 0.879i·2-s − 0.577i·3-s + 0.226·4-s + (0.853 + 0.521i)5-s + 0.507·6-s − 1.19i·7-s + 1.07i·8-s − 0.333·9-s + (−0.458 + 0.750i)10-s + 0.301·11-s − 0.130i·12-s + 1.92i·13-s + 1.05·14-s + (0.301 − 0.492i)15-s − 0.721·16-s − 0.753i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.521 - 0.853i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.521 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.521 - 0.853i$
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: \((2,\ 165,\ (\ :5/2),\ 0.521 - 0.853i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.765377024\)
\(L(\frac12)\) \(\approx\) \(2.765377024\)
\(L(\frac{7}{2})\) 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 \( 1 + (-47.6 - 29.1i)T \)
11 \( 1 - 121T \)
good2 \( 1 - 4.97iT - 32T^{2} \)
7 \( 1 + 155. iT - 1.68e4T^{2} \)
13 \( 1 - 1.17e3iT - 3.71e5T^{2} \)
17 \( 1 + 898. iT - 1.41e6T^{2} \)
19 \( 1 - 2.40e3T + 2.47e6T^{2} \)
23 \( 1 + 4.12e3iT - 6.43e6T^{2} \)
29 \( 1 + 75.4T + 2.05e7T^{2} \)
31 \( 1 - 3.59e3T + 2.86e7T^{2} \)
37 \( 1 - 1.60e4iT - 6.93e7T^{2} \)
41 \( 1 - 1.50e4T + 1.15e8T^{2} \)
43 \( 1 + 1.28e4iT - 1.47e8T^{2} \)
47 \( 1 - 6.69e3iT - 2.29e8T^{2} \)
53 \( 1 - 2.28e4iT - 4.18e8T^{2} \)
59 \( 1 - 1.30e4T + 7.14e8T^{2} \)
61 \( 1 - 882.T + 8.44e8T^{2} \)
67 \( 1 - 1.41e4iT - 1.35e9T^{2} \)
71 \( 1 + 7.87e4T + 1.80e9T^{2} \)
73 \( 1 - 3.42e4iT - 2.07e9T^{2} \)
79 \( 1 - 4.86e4T + 3.07e9T^{2} \)
83 \( 1 + 9.59e4iT - 3.93e9T^{2} \)
89 \( 1 + 1.34e3T + 5.58e9T^{2} \)
97 \( 1 + 8.09e4iT - 8.58e9T^{2} \)
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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

−11.95811668941065181966893969464, −11.18498105960298025120926771981, −10.02608856920349177156909523545, −8.871273819101402740737409587861, −7.38637628037054431918782119828, −6.89865943774439560226150959592, −6.12742701895393893877918984905, −4.61843424984776882729212725224, −2.67906572251731598196779484840, −1.30190252519175872162933659264, 1.05476032561846410147776699215, 2.44590284631555781559469529666, 3.48227165955088850853341655117, 5.35413659908393032951998553661, 5.93268537158360217928701210271, 7.81562816943982281304228946220, 9.179465495961864239380412952375, 9.787334758546139412102660393073, 10.72411718135937950480603827008, 11.77235301133037988946798563024

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