Properties

Label 2-15-15.8-c9-0-2
Degree $2$
Conductor $15$
Sign $-0.985 + 0.171i$
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  + (30.5 + 30.5i)2-s + (−129. − 54.5i)3-s + 1.35e3i·4-s + (−1.26e3 − 602. i)5-s + (−2.28e3 − 5.60e3i)6-s + (−2.74e3 + 2.74e3i)7-s + (−2.56e4 + 2.56e4i)8-s + (1.37e4 + 1.40e4i)9-s + (−2.01e4 − 5.68e4i)10-s − 4.81e3i·11-s + (7.36e4 − 1.74e5i)12-s + (6.67e4 + 6.67e4i)13-s − 1.67e5·14-s + (1.30e5 + 1.46e5i)15-s − 8.71e5·16-s + (3.46e4 + 3.46e4i)17-s + ⋯
L(s)  = 1  + (1.34 + 1.34i)2-s + (−0.921 − 0.388i)3-s + 2.63i·4-s + (−0.902 − 0.430i)5-s + (−0.718 − 1.76i)6-s + (−0.432 + 0.432i)7-s + (−2.21 + 2.21i)8-s + (0.698 + 0.715i)9-s + (−0.635 − 1.79i)10-s − 0.0991i·11-s + (1.02 − 2.43i)12-s + (0.648 + 0.648i)13-s − 1.16·14-s + (0.664 + 0.747i)15-s − 3.32·16-s + (0.100 + 0.100i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.171i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.985 + 0.171i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $-0.985 + 0.171i$
Analytic conductor: \(7.72553\)
Root analytic conductor: \(2.77948\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :9/2),\ -0.985 + 0.171i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.132163 - 1.52688i\)
\(L(\frac12)\) \(\approx\) \(0.132163 - 1.52688i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (129. + 54.5i)T \)
5 \( 1 + (1.26e3 + 602. i)T \)
good2 \( 1 + (-30.5 - 30.5i)T + 512iT^{2} \)
7 \( 1 + (2.74e3 - 2.74e3i)T - 4.03e7iT^{2} \)
11 \( 1 + 4.81e3iT - 2.35e9T^{2} \)
13 \( 1 + (-6.67e4 - 6.67e4i)T + 1.06e10iT^{2} \)
17 \( 1 + (-3.46e4 - 3.46e4i)T + 1.18e11iT^{2} \)
19 \( 1 - 4.27e5iT - 3.22e11T^{2} \)
23 \( 1 + (6.70e5 - 6.70e5i)T - 1.80e12iT^{2} \)
29 \( 1 + 8.83e5T + 1.45e13T^{2} \)
31 \( 1 - 7.22e6T + 2.64e13T^{2} \)
37 \( 1 + (5.43e6 - 5.43e6i)T - 1.29e14iT^{2} \)
41 \( 1 + 2.76e7iT - 3.27e14T^{2} \)
43 \( 1 + (1.38e7 + 1.38e7i)T + 5.02e14iT^{2} \)
47 \( 1 + (1.97e6 + 1.97e6i)T + 1.11e15iT^{2} \)
53 \( 1 + (-1.15e7 + 1.15e7i)T - 3.29e15iT^{2} \)
59 \( 1 + 2.00e7T + 8.66e15T^{2} \)
61 \( 1 - 1.28e8T + 1.16e16T^{2} \)
67 \( 1 + (9.09e7 - 9.09e7i)T - 2.72e16iT^{2} \)
71 \( 1 - 4.40e7iT - 4.58e16T^{2} \)
73 \( 1 + (-3.10e8 - 3.10e8i)T + 5.88e16iT^{2} \)
79 \( 1 - 3.03e8iT - 1.19e17T^{2} \)
83 \( 1 + (4.46e8 - 4.46e8i)T - 1.86e17iT^{2} \)
89 \( 1 - 1.06e9T + 3.50e17T^{2} \)
97 \( 1 + (7.40e8 - 7.40e8i)T - 7.60e17iT^{2} \)
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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

−17.19705648138105548850389285333, −16.18598398313784609953725642872, −15.50891163955098891147699759640, −13.75169048647412546761672495297, −12.50114891600751844868904197898, −11.70689839497678131094344395692, −8.237148201616436733649893462545, −6.80316733557740327155789335475, −5.51371868466814903933358804095, −3.97829937909448120232360699374, 0.59288851928716532929711517672, 3.34299702250280626020430318322, 4.62278214533908473450070609028, 6.38856078097919595558850424611, 10.10885863353521849253376331135, 11.06801914570780072584587317787, 12.05606404050965148992132102443, 13.27400505055870406278268423500, 14.94023815380526191066468351301, 15.97340045639437394292169053924

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