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Properties

Label 2-15-15.8-c11-0-13
Degree $2$
Conductor $15$
Sign $0.905 - 0.424i$
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  + (21.4 + 21.4i)2-s + (205. + 367. i)3-s − 1.12e3i·4-s + (6.18e3 − 3.24e3i)5-s + (−3.47e3 + 1.22e4i)6-s + (4.53e4 − 4.53e4i)7-s + (6.80e4 − 6.80e4i)8-s + (−9.29e4 + 1.50e5i)9-s + (2.02e5 + 6.31e4i)10-s + 6.45e5i·11-s + (4.14e5 − 2.31e5i)12-s + (−9.46e5 − 9.46e5i)13-s + 1.94e6·14-s + (2.46e6 + 1.60e6i)15-s + 6.06e5·16-s + (7.24e6 + 7.24e6i)17-s + ⋯
L(s)  = 1  + (0.473 + 0.473i)2-s + (0.487 + 0.873i)3-s − 0.551i·4-s + (0.885 − 0.464i)5-s + (−0.182 + 0.644i)6-s + (1.02 − 1.02i)7-s + (0.734 − 0.734i)8-s + (−0.524 + 0.851i)9-s + (0.639 + 0.199i)10-s + 1.20i·11-s + (0.481 − 0.268i)12-s + (−0.707 − 0.707i)13-s + 0.967·14-s + (0.837 + 0.546i)15-s + 0.144·16-s + (1.23 + 1.23i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.905 - 0.424i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.905 - 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.905 - 0.424i$
Analytic conductor: \(11.5251\)
Root analytic conductor: \(3.39487\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :11/2),\ 0.905 - 0.424i)\)

Particular Values

\(L(6)\) \(\approx\) \(3.16327 + 0.704826i\)
\(L(\frac12)\) \(\approx\) \(3.16327 + 0.704826i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-205. - 367. i)T \)
5 \( 1 + (-6.18e3 + 3.24e3i)T \)
good2 \( 1 + (-21.4 - 21.4i)T + 2.04e3iT^{2} \)
7 \( 1 + (-4.53e4 + 4.53e4i)T - 1.97e9iT^{2} \)
11 \( 1 - 6.45e5iT - 2.85e11T^{2} \)
13 \( 1 + (9.46e5 + 9.46e5i)T + 1.79e12iT^{2} \)
17 \( 1 + (-7.24e6 - 7.24e6i)T + 3.42e13iT^{2} \)
19 \( 1 + 4.63e6iT - 1.16e14T^{2} \)
23 \( 1 + (7.22e6 - 7.22e6i)T - 9.52e14iT^{2} \)
29 \( 1 + 7.09e7T + 1.22e16T^{2} \)
31 \( 1 + 1.10e8T + 2.54e16T^{2} \)
37 \( 1 + (-1.47e7 + 1.47e7i)T - 1.77e17iT^{2} \)
41 \( 1 - 5.09e8iT - 5.50e17T^{2} \)
43 \( 1 + (-9.36e8 - 9.36e8i)T + 9.29e17iT^{2} \)
47 \( 1 + (1.38e9 + 1.38e9i)T + 2.47e18iT^{2} \)
53 \( 1 + (-7.45e8 + 7.45e8i)T - 9.26e18iT^{2} \)
59 \( 1 + 3.69e9T + 3.01e19T^{2} \)
61 \( 1 + 4.33e8T + 4.35e19T^{2} \)
67 \( 1 + (1.02e10 - 1.02e10i)T - 1.22e20iT^{2} \)
71 \( 1 - 1.01e10iT - 2.31e20T^{2} \)
73 \( 1 + (8.86e9 + 8.86e9i)T + 3.13e20iT^{2} \)
79 \( 1 + 1.93e10iT - 7.47e20T^{2} \)
83 \( 1 + (1.92e10 - 1.92e10i)T - 1.28e21iT^{2} \)
89 \( 1 + 3.40e9T + 2.77e21T^{2} \)
97 \( 1 + (3.61e10 - 3.61e10i)T - 7.15e21iT^{2} \)
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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

−16.62760078485409687698525692358, −14.91672746943322067944474671517, −14.43388644603887586706263333980, −13.08588070767247879282285032216, −10.52487052644705116455524693779, −9.750648768297419440992212762952, −7.67832678130075683826673011560, −5.41265541367710098038177887017, −4.38458982107145573735206162656, −1.59410358331802555784145373347, 1.85704147132682025043524339925, 2.97057702037540945668580553588, 5.56334143935706263719237086640, 7.58487087413983405749257183688, 9.003795929190402211036607720205, 11.38500426114311716930412142001, 12.34791924098258770798076563845, 13.89521393945104978863266075531, 14.42465252895458263364848186685, 16.81389662427753229037741757239

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