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Properties

Label 2-15-15.2-c11-0-7
Degree $2$
Conductor $15$
Sign $0.561 - 0.827i$
Analytic cond. $11.5251$
Root an. cond. $3.39487$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (43.6 − 43.6i)2-s + (46.3 + 418. i)3-s − 1.75e3i·4-s + (−6.65e3 + 2.13e3i)5-s + (2.02e4 + 1.62e4i)6-s + (3.54e4 + 3.54e4i)7-s + (1.27e4 + 1.27e4i)8-s + (−1.72e5 + 3.87e4i)9-s + (−1.97e5 + 3.83e5i)10-s + 5.24e5i·11-s + (7.34e5 − 8.13e4i)12-s + (−4.09e5 + 4.09e5i)13-s + 3.08e6·14-s + (−1.20e6 − 2.68e6i)15-s + 4.70e6·16-s + (4.40e6 − 4.40e6i)17-s + ⋯
L(s)  = 1  + (0.963 − 0.963i)2-s + (0.110 + 0.993i)3-s − 0.857i·4-s + (−0.952 + 0.305i)5-s + (1.06 + 0.851i)6-s + (0.796 + 0.796i)7-s + (0.137 + 0.137i)8-s + (−0.975 + 0.218i)9-s + (−0.623 + 1.21i)10-s + 0.982i·11-s + (0.852 − 0.0944i)12-s + (−0.305 + 0.305i)13-s + 1.53·14-s + (−0.408 − 0.912i)15-s + 1.12·16-s + (0.753 − 0.753i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.561 - 0.827i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.561 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.561 - 0.827i$
Analytic conductor: \(11.5251\)
Root analytic conductor: \(3.39487\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :11/2),\ 0.561 - 0.827i)\)

Particular Values

\(L(6)\) \(\approx\) \(2.22903 + 1.18080i\)
\(L(\frac12)\) \(\approx\) \(2.22903 + 1.18080i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-46.3 - 418. i)T \)
5 \( 1 + (6.65e3 - 2.13e3i)T \)
good2 \( 1 + (-43.6 + 43.6i)T - 2.04e3iT^{2} \)
7 \( 1 + (-3.54e4 - 3.54e4i)T + 1.97e9iT^{2} \)
11 \( 1 - 5.24e5iT - 2.85e11T^{2} \)
13 \( 1 + (4.09e5 - 4.09e5i)T - 1.79e12iT^{2} \)
17 \( 1 + (-4.40e6 + 4.40e6i)T - 3.42e13iT^{2} \)
19 \( 1 - 9.09e6iT - 1.16e14T^{2} \)
23 \( 1 + (3.10e7 + 3.10e7i)T + 9.52e14iT^{2} \)
29 \( 1 - 1.53e7T + 1.22e16T^{2} \)
31 \( 1 + 1.36e8T + 2.54e16T^{2} \)
37 \( 1 + (-5.28e8 - 5.28e8i)T + 1.77e17iT^{2} \)
41 \( 1 + 1.17e8iT - 5.50e17T^{2} \)
43 \( 1 + (6.42e8 - 6.42e8i)T - 9.29e17iT^{2} \)
47 \( 1 + (-1.78e9 + 1.78e9i)T - 2.47e18iT^{2} \)
53 \( 1 + (-6.96e8 - 6.96e8i)T + 9.26e18iT^{2} \)
59 \( 1 - 2.53e9T + 3.01e19T^{2} \)
61 \( 1 - 1.20e10T + 4.35e19T^{2} \)
67 \( 1 + (-4.84e9 - 4.84e9i)T + 1.22e20iT^{2} \)
71 \( 1 + 1.33e10iT - 2.31e20T^{2} \)
73 \( 1 + (-1.02e10 + 1.02e10i)T - 3.13e20iT^{2} \)
79 \( 1 - 2.79e10iT - 7.47e20T^{2} \)
83 \( 1 + (6.76e9 + 6.76e9i)T + 1.28e21iT^{2} \)
89 \( 1 - 1.37e9T + 2.77e21T^{2} \)
97 \( 1 + (2.33e9 + 2.33e9i)T + 7.15e21iT^{2} \)
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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

−16.51448244138716594964812746050, −14.98767522684558893506643735080, −14.38019353471508065653749948874, −12.19540380427631768905081280100, −11.53695205885380128313337675866, −10.11559207550508772117557177661, −8.097679922374275802961645999622, −5.08734991596294074472399682377, −3.98499220653645578415854971969, −2.42914197039837283743952857533, 0.872900758934897597093506935902, 3.81996097083438400721754161354, 5.58598966615596850237944887569, 7.30955612172001195141774171073, 8.120083795772779796377058466438, 11.22262942323881608552154262034, 12.68407807565916103589934154915, 13.84532938448004476753830423043, 14.80069751199528324636162170056, 16.25392943927592937166785886531

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