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Properties

Label 2-15-15.14-c4-0-5
Degree $2$
Conductor $15$
Sign $-0.688 + 0.724i$
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  − 3.16·2-s + (−4.74 − 7.64i)3-s − 5.99·4-s + (−6.32 − 24.1i)5-s + (15.0 + 24.1i)6-s + 15.2i·7-s + 69.5·8-s + (−36.0 + 72.5i)9-s + (20.0 + 76.4i)10-s − 96.7i·11-s + (28.4 + 45.8i)12-s − 244. i·13-s − 48.3i·14-s + (−154. + 163. i)15-s − 124.·16-s − 278.·17-s + ⋯
L(s)  = 1  − 0.790·2-s + (−0.527 − 0.849i)3-s − 0.374·4-s + (−0.252 − 0.967i)5-s + (0.416 + 0.671i)6-s + 0.312i·7-s + 1.08·8-s + (−0.444 + 0.895i)9-s + (0.200 + 0.764i)10-s − 0.799i·11-s + (0.197 + 0.318i)12-s − 1.44i·13-s − 0.246i·14-s + (−0.688 + 0.724i)15-s − 0.484·16-s − 0.962·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.688 + 0.724i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.688 + 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $-0.688 + 0.724i$
Analytic conductor: \(1.55054\)
Root analytic conductor: \(1.24521\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :2),\ -0.688 + 0.724i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.182399 - 0.424953i\)
\(L(\frac12)\) \(\approx\) \(0.182399 - 0.424953i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.74 + 7.64i)T \)
5 \( 1 + (6.32 + 24.1i)T \)
good2 \( 1 + 3.16T + 16T^{2} \)
7 \( 1 - 15.2iT - 2.40e3T^{2} \)
11 \( 1 + 96.7iT - 1.46e4T^{2} \)
13 \( 1 + 244. iT - 2.85e4T^{2} \)
17 \( 1 + 278.T + 8.35e4T^{2} \)
19 \( 1 - 308T + 1.30e5T^{2} \)
23 \( 1 - 414.T + 2.79e5T^{2} \)
29 \( 1 - 193. iT - 7.07e5T^{2} \)
31 \( 1 - 32T + 9.23e5T^{2} \)
37 \( 1 + 1.28e3iT - 1.87e6T^{2} \)
41 \( 1 + 2.08e3iT - 2.82e6T^{2} \)
43 \( 1 - 2.58e3iT - 3.41e6T^{2} \)
47 \( 1 - 2.44e3T + 4.87e6T^{2} \)
53 \( 1 + 1.41e3T + 7.89e6T^{2} \)
59 \( 1 + 3.96e3iT - 1.21e7T^{2} \)
61 \( 1 + 928T + 1.38e7T^{2} \)
67 \( 1 + 2.58e3iT - 2.01e7T^{2} \)
71 \( 1 - 4.64e3iT - 2.54e7T^{2} \)
73 \( 1 + 4.22e3iT - 2.83e7T^{2} \)
79 \( 1 - 8T + 3.89e7T^{2} \)
83 \( 1 - 4.43e3T + 4.74e7T^{2} \)
89 \( 1 - 9.28e3iT - 6.27e7T^{2} \)
97 \( 1 + 2.50e3iT - 8.85e7T^{2} \)
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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

−18.02000509178939455781142196091, −17.18153782486423922907454987151, −15.94947426768237347976251240780, −13.58553822183746111780954719021, −12.60391530566199966930289373228, −10.93840579576637909271856101391, −8.934506284175130797282004890721, −7.78621412756573143209171849652, −5.34451405594234130299634987933, −0.67869032812669492665404954645, 4.36428104957663931477844254646, 7.01623531395304500488622008052, 9.183210795833782166499066918072, 10.33604359997966238932939510815, 11.55836504917335380857557717158, 13.90815546262867369401750147271, 15.30302731824090514383739566028, 16.71847711780028608314617255446, 17.76154618437813476227126249286, 18.81936232003463510747271730784

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