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Properties

Label 2-165-5.4-c5-0-9
Degree $2$
Conductor $165$
Sign $-0.987 - 0.155i$
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  + 10.7i·2-s − 9i·3-s − 83.6·4-s + (8.68 − 55.2i)5-s + 96.7·6-s + 41.3i·7-s − 555. i·8-s − 81·9-s + (593. + 93.3i)10-s + 121·11-s + 752. i·12-s + 13.6i·13-s − 444.·14-s + (−497. − 78.1i)15-s + 3.29e3·16-s + 1.14e3i·17-s + ⋯
L(s)  = 1  + 1.90i·2-s − 0.577i·3-s − 2.61·4-s + (0.155 − 0.987i)5-s + 1.09·6-s + 0.319i·7-s − 3.06i·8-s − 0.333·9-s + (1.87 + 0.295i)10-s + 0.301·11-s + 1.50i·12-s + 0.0224i·13-s − 0.606·14-s + (−0.570 − 0.0896i)15-s + 3.21·16-s + 0.960i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.155i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.987 - 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $-0.987 - 0.155i$
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: \((2,\ 165,\ (\ :5/2),\ -0.987 - 0.155i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.131105944\)
\(L(\frac12)\) \(\approx\) \(1.131105944\)
\(L(\frac{7}{2})\) 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 \( 1 + (-8.68 + 55.2i)T \)
11 \( 1 - 121T \)
good2 \( 1 - 10.7iT - 32T^{2} \)
7 \( 1 - 41.3iT - 1.68e4T^{2} \)
13 \( 1 - 13.6iT - 3.71e5T^{2} \)
17 \( 1 - 1.14e3iT - 1.41e6T^{2} \)
19 \( 1 - 1.42e3T + 2.47e6T^{2} \)
23 \( 1 - 2.53e3iT - 6.43e6T^{2} \)
29 \( 1 + 8.59e3T + 2.05e7T^{2} \)
31 \( 1 - 434.T + 2.86e7T^{2} \)
37 \( 1 - 1.44e4iT - 6.93e7T^{2} \)
41 \( 1 - 1.23e4T + 1.15e8T^{2} \)
43 \( 1 - 2.69e3iT - 1.47e8T^{2} \)
47 \( 1 + 1.69e3iT - 2.29e8T^{2} \)
53 \( 1 - 9.46e3iT - 4.18e8T^{2} \)
59 \( 1 - 4.14e4T + 7.14e8T^{2} \)
61 \( 1 + 1.78e4T + 8.44e8T^{2} \)
67 \( 1 - 5.17e4iT - 1.35e9T^{2} \)
71 \( 1 + 1.66e4T + 1.80e9T^{2} \)
73 \( 1 - 4.97e4iT - 2.07e9T^{2} \)
79 \( 1 + 7.14e4T + 3.07e9T^{2} \)
83 \( 1 - 3.35e4iT - 3.93e9T^{2} \)
89 \( 1 - 7.69e4T + 5.58e9T^{2} \)
97 \( 1 - 1.77e4iT - 8.58e9T^{2} \)
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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

−12.95084283189914793835355834735, −11.77889794139287545262111133593, −9.719258899363560017987640740858, −8.931909939436429029320188343209, −8.074653807766877112420754257333, −7.22145029243102157105344864208, −5.95492743794028727421864331494, −5.36020185164972350547935547455, −3.99710874279515534219403575974, −1.19485169329544776533358218883, 0.43049109041887096465169198765, 2.18114950397577632950233328896, 3.26139652146262805849512362788, 4.18528703945670750630216258925, 5.55154318665161942933940123573, 7.45791770646820351679425530293, 9.085970407996705668848592337726, 9.712254403673748271324589444116, 10.67990431350579223791344759999, 11.20897460486317560322993053075

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