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Properties

Label 2-15-15.14-c6-0-4
Degree $2$
Conductor $15$
Sign $-0.259 + 0.965i$
Analytic cond. $3.45081$
Root an. cond. $1.85763$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 11.8·2-s + (9.31 + 25.3i)3-s + 77.4·4-s + (−102. − 72.0i)5-s + (−110. − 301. i)6-s − 553. i·7-s − 160.·8-s + (−555. + 471. i)9-s + (1.21e3 + 857. i)10-s − 799. i·11-s + (721. + 1.96e3i)12-s − 1.11e3i·13-s + 6.57e3i·14-s + (875. − 3.25e3i)15-s − 3.05e3·16-s − 2.58e3·17-s + ⋯
L(s)  = 1  − 1.48·2-s + (0.344 + 0.938i)3-s + 1.21·4-s + (−0.817 − 0.576i)5-s + (−0.512 − 1.39i)6-s − 1.61i·7-s − 0.313·8-s + (−0.762 + 0.647i)9-s + (1.21 + 0.857i)10-s − 0.600i·11-s + (0.417 + 1.13i)12-s − 0.509i·13-s + 2.39i·14-s + (0.259 − 0.965i)15-s − 0.745·16-s − 0.525·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.259 + 0.965i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.259 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $-0.259 + 0.965i$
Analytic conductor: \(3.45081\)
Root analytic conductor: \(1.85763\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :3),\ -0.259 + 0.965i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.220236 - 0.287227i\)
\(L(\frac12)\) \(\approx\) \(0.220236 - 0.287227i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-9.31 - 25.3i)T \)
5 \( 1 + (102. + 72.0i)T \)
good2 \( 1 + 11.8T + 64T^{2} \)
7 \( 1 + 553. iT - 1.17e5T^{2} \)
11 \( 1 + 799. iT - 1.77e6T^{2} \)
13 \( 1 + 1.11e3iT - 4.82e6T^{2} \)
17 \( 1 + 2.58e3T + 2.41e7T^{2} \)
19 \( 1 + 3.49e3T + 4.70e7T^{2} \)
23 \( 1 + 5.05e3T + 1.48e8T^{2} \)
29 \( 1 + 2.54e4iT - 5.94e8T^{2} \)
31 \( 1 + 1.19e4T + 8.87e8T^{2} \)
37 \( 1 - 2.87e4iT - 2.56e9T^{2} \)
41 \( 1 + 5.56e4iT - 4.75e9T^{2} \)
43 \( 1 + 3.85e4iT - 6.32e9T^{2} \)
47 \( 1 - 1.51e5T + 1.07e10T^{2} \)
53 \( 1 + 1.43e5T + 2.21e10T^{2} \)
59 \( 1 - 4.01e5iT - 4.21e10T^{2} \)
61 \( 1 - 3.09e5T + 5.15e10T^{2} \)
67 \( 1 + 1.63e5iT - 9.04e10T^{2} \)
71 \( 1 + 1.51e5iT - 1.28e11T^{2} \)
73 \( 1 - 7.01e4iT - 1.51e11T^{2} \)
79 \( 1 - 4.37e5T + 2.43e11T^{2} \)
83 \( 1 + 3.42e5T + 3.26e11T^{2} \)
89 \( 1 + 6.31e5iT - 4.96e11T^{2} \)
97 \( 1 + 1.55e6iT - 8.32e11T^{2} \)
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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

−17.28493772015846078490906483237, −16.59063409306786097161578709235, −15.55499653027575315169709811352, −13.65175433346085459751014733861, −11.14123179908577220261444675541, −10.21521008988746375768708899258, −8.716113918729575452171918257266, −7.61754281234328233536766956929, −4.13729530574246336999612599256, −0.39688891356929880813847369461, 2.21846057663547597741491166243, 6.74642369724622901290048741803, 8.138576322406231321037058296845, 9.192474399682357279707026093986, 11.26076128841951418920706176990, 12.43956873530696539464321167147, 14.67738792731770494622746162312, 15.89497731027171350591601195088, 17.69848328636429356840529723846, 18.53870397693922513861174582265

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