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Properties

Label 2-15-5.4-c9-0-7
Degree $2$
Conductor $15$
Sign $-0.319 - 0.947i$
Analytic cond. $7.72553$
Root an. cond. $2.77948$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 33.3i·2-s − 81i·3-s − 602.·4-s + (−1.32e3 + 445. i)5-s − 2.70e3·6-s + 176. i·7-s + 3.02e3i·8-s − 6.56e3·9-s + (1.48e4 + 4.42e4i)10-s + 7.38e3·11-s + 4.88e4i·12-s + 1.04e5i·13-s + 5.87e3·14-s + (3.61e4 + 1.07e5i)15-s − 2.07e5·16-s − 5.11e5i·17-s + ⋯
L(s)  = 1  − 1.47i·2-s − 0.577i·3-s − 1.17·4-s + (−0.947 + 0.319i)5-s − 0.851·6-s + 0.0277i·7-s + 0.260i·8-s − 0.333·9-s + (0.470 + 1.39i)10-s + 0.152·11-s + 0.679i·12-s + 1.01i·13-s + 0.0409·14-s + (0.184 + 0.547i)15-s − 0.791·16-s − 1.48i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.319 - 0.947i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.319 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $-0.319 - 0.947i$
Analytic conductor: \(7.72553\)
Root analytic conductor: \(2.77948\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :9/2),\ -0.319 - 0.947i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.358692 + 0.499225i\)
\(L(\frac12)\) \(\approx\) \(0.358692 + 0.499225i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 81iT \)
5 \( 1 + (1.32e3 - 445. i)T \)
good2 \( 1 + 33.3iT - 512T^{2} \)
7 \( 1 - 176. iT - 4.03e7T^{2} \)
11 \( 1 - 7.38e3T + 2.35e9T^{2} \)
13 \( 1 - 1.04e5iT - 1.06e10T^{2} \)
17 \( 1 + 5.11e5iT - 1.18e11T^{2} \)
19 \( 1 + 8.33e5T + 3.22e11T^{2} \)
23 \( 1 + 8.68e5iT - 1.80e12T^{2} \)
29 \( 1 + 5.35e6T + 1.45e13T^{2} \)
31 \( 1 + 4.02e6T + 2.64e13T^{2} \)
37 \( 1 + 1.22e7iT - 1.29e14T^{2} \)
41 \( 1 - 3.24e7T + 3.27e14T^{2} \)
43 \( 1 - 1.40e7iT - 5.02e14T^{2} \)
47 \( 1 + 4.74e7iT - 1.11e15T^{2} \)
53 \( 1 + 4.00e7iT - 3.29e15T^{2} \)
59 \( 1 - 1.14e8T + 8.66e15T^{2} \)
61 \( 1 + 4.22e7T + 1.16e16T^{2} \)
67 \( 1 + 1.12e8iT - 2.72e16T^{2} \)
71 \( 1 + 2.94e8T + 4.58e16T^{2} \)
73 \( 1 - 2.45e8iT - 5.88e16T^{2} \)
79 \( 1 - 4.95e7T + 1.19e17T^{2} \)
83 \( 1 + 1.96e8iT - 1.86e17T^{2} \)
89 \( 1 + 1.02e8T + 3.50e17T^{2} \)
97 \( 1 + 1.08e8iT - 7.60e17T^{2} \)
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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

−16.34793636709154656402675884552, −14.49801181459105461546797860155, −12.92980770506339202462585602875, −11.80048284822693814495755260665, −10.94026197155017017849361623418, −9.060653184463443473028689818546, −7.07278249468186176882932944629, −4.06778738609871268909513987400, −2.30448770745738778793287466997, −0.31333241160882256906953373122, 4.12379854594605978594935926983, 5.80958119455002037402999776781, 7.67094126363671928906702221481, 8.777670813381036487414965566121, 10.90704828485829493748976033736, 12.85936310391525868711308885759, 14.83558067418063337522674439389, 15.34496387767449226786306655078, 16.57192845120371492423612257097, 17.44604962275465401621037264173

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