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Properties

Label 2-15-3.2-c4-0-3
Degree $2$
Conductor $15$
Sign $0.934 + 0.356i$
Analytic cond. $1.55054$
Root an. cond. $1.24521$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.407i·2-s + (3.21 − 8.40i)3-s + 15.8·4-s + 11.1i·5-s + (3.42 + 1.30i)6-s − 46.9·7-s + 12.9i·8-s + (−60.3 − 54.0i)9-s − 4.55·10-s + 200. i·11-s + (50.8 − 133. i)12-s − 22.3·13-s − 19.1i·14-s + (93.9 + 35.9i)15-s + 248.·16-s − 344. i·17-s + ⋯
L(s)  = 1  + 0.101i·2-s + (0.356 − 0.934i)3-s + 0.989·4-s + 0.447i·5-s + (0.0951 + 0.0363i)6-s − 0.958·7-s + 0.202i·8-s + (−0.745 − 0.666i)9-s − 0.0455·10-s + 1.66i·11-s + (0.353 − 0.924i)12-s − 0.132·13-s − 0.0976i·14-s + (0.417 + 0.159i)15-s + 0.968·16-s − 1.19i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.934 + 0.356i$
Analytic conductor: \(1.55054\)
Root analytic conductor: \(1.24521\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :2),\ 0.934 + 0.356i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.34539 - 0.248332i\)
\(L(\frac12)\) \(\approx\) \(1.34539 - 0.248332i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-3.21 + 8.40i)T \)
5 \( 1 - 11.1iT \)
good2 \( 1 - 0.407iT - 16T^{2} \)
7 \( 1 + 46.9T + 2.40e3T^{2} \)
11 \( 1 - 200. iT - 1.46e4T^{2} \)
13 \( 1 + 22.3T + 2.85e4T^{2} \)
17 \( 1 + 344. iT - 8.35e4T^{2} \)
19 \( 1 + 59.9T + 1.30e5T^{2} \)
23 \( 1 + 212. iT - 2.79e5T^{2} \)
29 \( 1 + 578. iT - 7.07e5T^{2} \)
31 \( 1 - 490.T + 9.23e5T^{2} \)
37 \( 1 - 1.93e3T + 1.87e6T^{2} \)
41 \( 1 + 1.63e3iT - 2.82e6T^{2} \)
43 \( 1 + 2.16e3T + 3.41e6T^{2} \)
47 \( 1 - 2.28e3iT - 4.87e6T^{2} \)
53 \( 1 - 2.62e3iT - 7.89e6T^{2} \)
59 \( 1 + 4.10e3iT - 1.21e7T^{2} \)
61 \( 1 - 4.79e3T + 1.38e7T^{2} \)
67 \( 1 + 1.56e3T + 2.01e7T^{2} \)
71 \( 1 - 5.37e3iT - 2.54e7T^{2} \)
73 \( 1 + 4.32e3T + 2.83e7T^{2} \)
79 \( 1 + 4.80e3T + 3.89e7T^{2} \)
83 \( 1 - 3.38e3iT - 4.74e7T^{2} \)
89 \( 1 + 3.44e3iT - 6.27e7T^{2} \)
97 \( 1 - 5.44e3T + 8.85e7T^{2} \)
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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

−18.68117783713887414421793212428, −17.32363108172737280342234537481, −15.72912364192070050163525685770, −14.55454217414042127156110981680, −12.87361336814546512882252654468, −11.75073584623664167172399111551, −9.808552419872132758236017975395, −7.46947561328559555476887595680, −6.51769906653217427241821928569, −2.55489415175645136772515103723, 3.32535912491251712982761644297, 6.02178243696339248007025197281, 8.436643640357143837407988803869, 10.10374022320347153386086607773, 11.39503695829434520259694211871, 13.19752140484046965050447663004, 14.93740210509966171151680193720, 16.20123018597429768200539386125, 16.69561070445604258284478337135, 19.26452200050636054926350348917

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