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Properties

Label 2-15-3.2-c4-0-1
Degree $2$
Conductor $15$
Sign $-0.224 - 0.974i$
Analytic cond. $1.55054$
Root an. cond. $1.24521$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7.20i·2-s + (8.77 − 2.01i)3-s − 35.9·4-s − 11.1i·5-s + (14.5 + 63.2i)6-s + 23.3·7-s − 144. i·8-s + (72.8 − 35.3i)9-s + 80.6·10-s + 33.9i·11-s + (−315. + 72.5i)12-s − 125.·13-s + 168. i·14-s + (−22.5 − 98.0i)15-s + 463.·16-s − 308. i·17-s + ⋯
L(s)  = 1  + 1.80i·2-s + (0.974 − 0.224i)3-s − 2.24·4-s − 0.447i·5-s + (0.403 + 1.75i)6-s + 0.476·7-s − 2.25i·8-s + (0.899 − 0.436i)9-s + 0.806·10-s + 0.280i·11-s + (−2.19 + 0.503i)12-s − 0.745·13-s + 0.858i·14-s + (−0.100 − 0.435i)15-s + 1.80·16-s − 1.06i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.224 - 0.974i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.224 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $-0.224 - 0.974i$
Analytic conductor: \(1.55054\)
Root analytic conductor: \(1.24521\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :2),\ -0.224 - 0.974i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.843993 + 1.06010i\)
\(L(\frac12)\) \(\approx\) \(0.843993 + 1.06010i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-8.77 + 2.01i)T \)
5 \( 1 + 11.1iT \)
good2 \( 1 - 7.20iT - 16T^{2} \)
7 \( 1 - 23.3T + 2.40e3T^{2} \)
11 \( 1 - 33.9iT - 1.46e4T^{2} \)
13 \( 1 + 125.T + 2.85e4T^{2} \)
17 \( 1 + 308. iT - 8.35e4T^{2} \)
19 \( 1 + 363.T + 1.30e5T^{2} \)
23 \( 1 - 102. iT - 2.79e5T^{2} \)
29 \( 1 - 1.56e3iT - 7.07e5T^{2} \)
31 \( 1 - 326.T + 9.23e5T^{2} \)
37 \( 1 - 72.4T + 1.87e6T^{2} \)
41 \( 1 - 2.14e3iT - 2.82e6T^{2} \)
43 \( 1 - 1.50e3T + 3.41e6T^{2} \)
47 \( 1 + 1.95e3iT - 4.87e6T^{2} \)
53 \( 1 - 2.96e3iT - 7.89e6T^{2} \)
59 \( 1 + 5.04e3iT - 1.21e7T^{2} \)
61 \( 1 - 736.T + 1.38e7T^{2} \)
67 \( 1 - 4.25e3T + 2.01e7T^{2} \)
71 \( 1 - 1.30e3iT - 2.54e7T^{2} \)
73 \( 1 - 6.76e3T + 2.83e7T^{2} \)
79 \( 1 + 6.36e3T + 3.89e7T^{2} \)
83 \( 1 + 6.89e3iT - 4.74e7T^{2} \)
89 \( 1 + 6.40e3iT - 6.27e7T^{2} \)
97 \( 1 + 9.87e3T + 8.85e7T^{2} \)
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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

−18.56949804782934623750602522410, −17.44491016516588968870837324696, −16.13395925392414341930110194390, −14.94108742207748997698544402298, −14.10535274401709028493605713906, −12.76998990084292776508444807808, −9.428111401023613053448076948622, −8.234456067188329212862238679530, −7.01388605599170706138399106500, −4.78395746241869307537520082550, 2.31137887005096632444979580234, 4.12723968844093698482789691866, 8.314438726321527638568650041886, 9.830130373738813280646287040215, 10.94255400681828369255585473219, 12.54913437188157948880901370100, 13.83614204849502010616700702928, 14.98433454781507841320993160168, 17.49104715848186168241008250092, 19.02360978357746591755202550050

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