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Properties

Label 2-15-5.4-c9-0-1
Degree $2$
Conductor $15$
Sign $-0.846 - 0.532i$
Analytic cond. $7.72553$
Root an. cond. $2.77948$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 20.9i·2-s − 81i·3-s + 72.2·4-s + (−743. + 1.18e3i)5-s + 1.69e3·6-s + 3.57e3i·7-s + 1.22e4i·8-s − 6.56e3·9-s + (−2.48e4 − 1.56e4i)10-s − 7.45e4·11-s − 5.85e3i·12-s + 3.44e4i·13-s − 7.49e4·14-s + (9.58e4 + 6.02e4i)15-s − 2.19e5·16-s + 3.72e5i·17-s + ⋯
L(s)  = 1  + 0.926i·2-s − 0.577i·3-s + 0.141·4-s + (−0.532 + 0.846i)5-s + 0.535·6-s + 0.562i·7-s + 1.05i·8-s − 0.333·9-s + (−0.784 − 0.493i)10-s − 1.53·11-s − 0.0814i·12-s + 0.334i·13-s − 0.521·14-s + (0.488 + 0.307i)15-s − 0.838·16-s + 1.08i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.846 - 0.532i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.846 - 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $-0.846 - 0.532i$
Analytic conductor: \(7.72553\)
Root analytic conductor: \(2.77948\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :9/2),\ -0.846 - 0.532i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.357843 + 1.24129i\)
\(L(\frac12)\) \(\approx\) \(0.357843 + 1.24129i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 81iT \)
5 \( 1 + (743. - 1.18e3i)T \)
good2 \( 1 - 20.9iT - 512T^{2} \)
7 \( 1 - 3.57e3iT - 4.03e7T^{2} \)
11 \( 1 + 7.45e4T + 2.35e9T^{2} \)
13 \( 1 - 3.44e4iT - 1.06e10T^{2} \)
17 \( 1 - 3.72e5iT - 1.18e11T^{2} \)
19 \( 1 - 8.35e5T + 3.22e11T^{2} \)
23 \( 1 + 3.18e4iT - 1.80e12T^{2} \)
29 \( 1 - 6.73e6T + 1.45e13T^{2} \)
31 \( 1 + 4.04e6T + 2.64e13T^{2} \)
37 \( 1 + 6.29e6iT - 1.29e14T^{2} \)
41 \( 1 - 1.10e7T + 3.27e14T^{2} \)
43 \( 1 + 1.42e7iT - 5.02e14T^{2} \)
47 \( 1 + 2.76e7iT - 1.11e15T^{2} \)
53 \( 1 - 8.30e7iT - 3.29e15T^{2} \)
59 \( 1 + 1.04e8T + 8.66e15T^{2} \)
61 \( 1 - 3.99e7T + 1.16e16T^{2} \)
67 \( 1 - 1.98e8iT - 2.72e16T^{2} \)
71 \( 1 - 4.52e7T + 4.58e16T^{2} \)
73 \( 1 - 3.64e8iT - 5.88e16T^{2} \)
79 \( 1 - 4.55e8T + 1.19e17T^{2} \)
83 \( 1 - 3.16e7iT - 1.86e17T^{2} \)
89 \( 1 + 2.77e8T + 3.50e17T^{2} \)
97 \( 1 - 1.06e9iT - 7.60e17T^{2} \)
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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

−17.78612189684622660363141659976, −16.06513688241723814125234340409, −15.24312302038325520629874552654, −13.96857516344532724439905914503, −12.14202356358743849494242312208, −10.74796321764520152480108107685, −8.199615545196421185780102426422, −7.15570224127282995710543137673, −5.67756715236357763390301332935, −2.60369591536547005127570239773, 0.66180733059183723265634497397, 3.09078466691506456604579187664, 4.91188250978158227800992380718, 7.68844737254465174524316455837, 9.674030031942456206357792689019, 10.90286146347150729309828582279, 12.16429121240798944759334747262, 13.49834155892655050150554472883, 15.70781906802098031285582491816, 16.24845752749723878794992213095

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