Properties

Label 2-15-3.2-c8-0-7
Degree $2$
Conductor $15$
Sign $-0.929 + 0.368i$
Analytic cond. $6.11067$
Root an. cond. $2.47197$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7.97i·2-s + (−29.8 − 75.3i)3-s + 192.·4-s − 279. i·5-s + (−600. + 238. i)6-s − 3.21e3·7-s − 3.57e3i·8-s + (−4.77e3 + 4.49e3i)9-s − 2.22e3·10-s − 6.48e3i·11-s + (−5.74e3 − 1.44e4i)12-s − 1.05e4·13-s + 2.56e4i·14-s + (−2.10e4 + 8.34e3i)15-s + 2.07e4·16-s − 5.77e4i·17-s + ⋯
L(s)  = 1  − 0.498i·2-s + (−0.368 − 0.929i)3-s + 0.751·4-s − 0.447i·5-s + (−0.463 + 0.183i)6-s − 1.33·7-s − 0.873i·8-s + (−0.728 + 0.685i)9-s − 0.222·10-s − 0.442i·11-s + (−0.276 − 0.698i)12-s − 0.367·13-s + 0.666i·14-s + (−0.415 + 0.164i)15-s + 0.316·16-s − 0.691i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.368i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $-0.929 + 0.368i$
Analytic conductor: \(6.11067\)
Root analytic conductor: \(2.47197\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :4),\ -0.929 + 0.368i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.232765 - 1.21913i\)
\(L(\frac12)\) \(\approx\) \(0.232765 - 1.21913i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (29.8 + 75.3i)T \)
5 \( 1 + 279. iT \)
good2 \( 1 + 7.97iT - 256T^{2} \)
7 \( 1 + 3.21e3T + 5.76e6T^{2} \)
11 \( 1 + 6.48e3iT - 2.14e8T^{2} \)
13 \( 1 + 1.05e4T + 8.15e8T^{2} \)
17 \( 1 + 5.77e4iT - 6.97e9T^{2} \)
19 \( 1 - 2.28e5T + 1.69e10T^{2} \)
23 \( 1 + 1.13e5iT - 7.83e10T^{2} \)
29 \( 1 + 1.11e6iT - 5.00e11T^{2} \)
31 \( 1 + 3.04e5T + 8.52e11T^{2} \)
37 \( 1 - 6.30e5T + 3.51e12T^{2} \)
41 \( 1 - 4.62e6iT - 7.98e12T^{2} \)
43 \( 1 - 5.44e6T + 1.16e13T^{2} \)
47 \( 1 + 6.69e6iT - 2.38e13T^{2} \)
53 \( 1 - 1.25e7iT - 6.22e13T^{2} \)
59 \( 1 + 5.55e6iT - 1.46e14T^{2} \)
61 \( 1 + 1.31e7T + 1.91e14T^{2} \)
67 \( 1 - 1.70e7T + 4.06e14T^{2} \)
71 \( 1 - 1.08e7iT - 6.45e14T^{2} \)
73 \( 1 + 2.67e7T + 8.06e14T^{2} \)
79 \( 1 + 3.07e6T + 1.51e15T^{2} \)
83 \( 1 - 2.32e7iT - 2.25e15T^{2} \)
89 \( 1 + 3.21e7iT - 3.93e15T^{2} \)
97 \( 1 + 9.03e7T + 7.83e15T^{2} \)
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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

−16.74478965533668506437249166374, −15.89424391217185138115799881684, −13.60070364893294022502598216928, −12.47485900980346579595489382644, −11.48560470928764313896343864900, −9.707047727282382733503702499170, −7.41226731040701118577275430996, −6.01823772211691002334916218218, −2.82421968919769239414377604649, −0.73022091392106362792043838316, 3.21185554214719111115494193287, 5.70256804534303582321523660741, 7.11088813044127468095931716105, 9.516044660351828082028459067360, 10.77563185432832503056698321154, 12.26550332971909960643651429049, 14.44159984518710515771193985858, 15.67586846669319658346207244030, 16.31843694685913653933662011843, 17.62410282239631123780943783221

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