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Properties

Label 2-15-15.14-c10-0-5
Degree $2$
Conductor $15$
Sign $-0.238 - 0.971i$
Analytic cond. $9.53035$
Root an. cond. $3.08712$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 36.7·2-s + (151. + 190. i)3-s + 324.·4-s + (2.83e3 + 1.30e3i)5-s + (−5.55e3 − 6.98e3i)6-s − 1.71e4i·7-s + 2.56e4·8-s + (−1.32e4 + 5.75e4i)9-s + (−1.04e5 − 4.79e4i)10-s + 5.77e4i·11-s + (4.91e4 + 6.17e4i)12-s + 7.26e5i·13-s + 6.31e5i·14-s + (1.81e5 + 7.37e5i)15-s − 1.27e6·16-s + 8.53e5·17-s + ⋯
L(s)  = 1  − 1.14·2-s + (0.622 + 0.782i)3-s + 0.317·4-s + (0.908 + 0.417i)5-s + (−0.714 − 0.897i)6-s − 1.02i·7-s + 0.783·8-s + (−0.224 + 0.974i)9-s + (−1.04 − 0.479i)10-s + 0.358i·11-s + (0.197 + 0.248i)12-s + 1.95i·13-s + 1.17i·14-s + (0.238 + 0.971i)15-s − 1.21·16-s + 0.601·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.238 - 0.971i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.238 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $-0.238 - 0.971i$
Analytic conductor: \(9.53035\)
Root analytic conductor: \(3.08712\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :5),\ -0.238 - 0.971i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.692626 + 0.883740i\)
\(L(\frac12)\) \(\approx\) \(0.692626 + 0.883740i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-151. - 190. i)T \)
5 \( 1 + (-2.83e3 - 1.30e3i)T \)
good2 \( 1 + 36.7T + 1.02e3T^{2} \)
7 \( 1 + 1.71e4iT - 2.82e8T^{2} \)
11 \( 1 - 5.77e4iT - 2.59e10T^{2} \)
13 \( 1 - 7.26e5iT - 1.37e11T^{2} \)
17 \( 1 - 8.53e5T + 2.01e12T^{2} \)
19 \( 1 + 3.02e6T + 6.13e12T^{2} \)
23 \( 1 + 1.73e6T + 4.14e13T^{2} \)
29 \( 1 - 2.25e7iT - 4.20e14T^{2} \)
31 \( 1 - 1.06e7T + 8.19e14T^{2} \)
37 \( 1 + 1.97e7iT - 4.80e15T^{2} \)
41 \( 1 - 8.40e7iT - 1.34e16T^{2} \)
43 \( 1 + 7.92e7iT - 2.16e16T^{2} \)
47 \( 1 - 1.87e8T + 5.25e16T^{2} \)
53 \( 1 - 5.59e8T + 1.74e17T^{2} \)
59 \( 1 + 6.70e8iT - 5.11e17T^{2} \)
61 \( 1 + 7.73e8T + 7.13e17T^{2} \)
67 \( 1 - 1.72e9iT - 1.82e18T^{2} \)
71 \( 1 - 9.90e8iT - 3.25e18T^{2} \)
73 \( 1 + 3.12e9iT - 4.29e18T^{2} \)
79 \( 1 - 1.32e9T + 9.46e18T^{2} \)
83 \( 1 + 1.92e9T + 1.55e19T^{2} \)
89 \( 1 + 1.80e9iT - 3.11e19T^{2} \)
97 \( 1 + 7.72e8iT - 7.37e19T^{2} \)
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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

−17.11900573417884697515201759037, −16.49903680434625914969706348698, −14.46197139845195422969127564669, −13.63153768209141192723126163259, −10.78640987289011833819971775952, −9.896692604550066750751588685414, −8.853209748399425502914934991220, −7.08823694687165371525248084806, −4.31274064327918055716854265919, −1.83374068769705291295674217646, 0.76109068779145001017583423818, 2.37940931238774668012135967731, 5.85336028200242062555995812924, 8.018499578667983199548736681880, 8.886930823041108145209632209600, 10.21438584138277311075898968387, 12.49559519044914014204597518875, 13.58466898572817730613809150159, 15.23099015110752165081769099563, 17.10633072127940794460247615610

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