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Properties

Label 2-15-5.4-c17-0-2
Degree $2$
Conductor $15$
Sign $0.873 + 0.485i$
Analytic cond. $27.4833$
Root an. cond. $5.24245$
Motivic weight $17$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 667. i·2-s − 6.56e3i·3-s − 3.14e5·4-s + (4.24e5 − 7.63e5i)5-s − 4.37e6·6-s + 1.80e7i·7-s + 1.22e8i·8-s − 4.30e7·9-s + (−5.09e8 − 2.83e8i)10-s + 4.36e8·11-s + 2.06e9i·12-s + 3.63e9i·13-s + 1.20e10·14-s + (−5.00e9 − 2.78e9i)15-s + 4.04e10·16-s + 3.85e10i·17-s + ⋯
L(s)  = 1  − 1.84i·2-s − 0.577i·3-s − 2.39·4-s + (0.485 − 0.873i)5-s − 1.06·6-s + 1.18i·7-s + 2.57i·8-s − 0.333·9-s + (−1.61 − 0.895i)10-s + 0.614·11-s + 1.38i·12-s + 1.23i·13-s + 2.18·14-s + (−0.504 − 0.280i)15-s + 2.35·16-s + 1.34i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.873 + 0.485i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.873 + 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.873 + 0.485i$
Analytic conductor: \(27.4833\)
Root analytic conductor: \(5.24245\)
Motivic weight: \(17\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :17/2),\ 0.873 + 0.485i)\)

Particular Values

\(L(9)\) \(\approx\) \(0.8375428190\)
\(L(\frac12)\) \(\approx\) \(0.8375428190\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 6.56e3iT \)
5 \( 1 + (-4.24e5 + 7.63e5i)T \)
good2 \( 1 + 667. iT - 1.31e5T^{2} \)
7 \( 1 - 1.80e7iT - 2.32e14T^{2} \)
11 \( 1 - 4.36e8T + 5.05e17T^{2} \)
13 \( 1 - 3.63e9iT - 8.65e18T^{2} \)
17 \( 1 - 3.85e10iT - 8.27e20T^{2} \)
19 \( 1 + 3.76e10T + 5.48e21T^{2} \)
23 \( 1 - 2.00e11iT - 1.41e23T^{2} \)
29 \( 1 + 2.22e12T + 7.25e24T^{2} \)
31 \( 1 + 8.21e12T + 2.25e25T^{2} \)
37 \( 1 + 3.23e13iT - 4.56e26T^{2} \)
41 \( 1 + 8.93e12T + 2.61e27T^{2} \)
43 \( 1 - 1.43e14iT - 5.87e27T^{2} \)
47 \( 1 + 1.91e14iT - 2.66e28T^{2} \)
53 \( 1 - 5.95e14iT - 2.05e29T^{2} \)
59 \( 1 + 7.42e14T + 1.27e30T^{2} \)
61 \( 1 - 4.42e13T + 2.24e30T^{2} \)
67 \( 1 - 2.86e15iT - 1.10e31T^{2} \)
71 \( 1 - 5.10e15T + 2.96e31T^{2} \)
73 \( 1 - 1.33e16iT - 4.74e31T^{2} \)
79 \( 1 + 4.70e14T + 1.81e32T^{2} \)
83 \( 1 + 1.35e16iT - 4.21e32T^{2} \)
89 \( 1 - 1.63e16T + 1.37e33T^{2} \)
97 \( 1 + 6.32e16iT - 5.95e33T^{2} \)
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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

−14.43391092684470472624538546669, −13.01099625278450561851557843856, −12.33088863052457300566903006721, −11.25060849288859262787082396931, −9.386897338852089446023160885128, −8.714597730104753388957277441531, −5.70699708944196965933568664261, −4.02549611387865764047946093867, −2.10652970047788831650409141214, −1.50317580828658809134976568632, 0.27710106551089450685524326123, 3.64033830320885403912846976760, 5.18641411457217587225153874835, 6.59804477616623462369605167328, 7.63389366535563111888399223389, 9.355399704015464531334518080931, 10.60074869142431642447289332015, 13.41588788696089403855292698674, 14.35296641761122906470293596006, 15.24335469655298023276674817479

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