Properties

Label 2-15-5.4-c7-0-6
Degree $2$
Conductor $15$
Sign $-0.978 - 0.205i$
Analytic cond. $4.68577$
Root an. cond. $2.16466$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 17.7i·2-s − 27i·3-s − 188.·4-s + (57.4 − 273. i)5-s − 480.·6-s + 1.23e3i·7-s + 1.06e3i·8-s − 729·9-s + (−4.86e3 − 1.02e3i)10-s + 1.21e3·11-s + 5.08e3i·12-s − 1.35e4i·13-s + 2.19e4·14-s + (−7.38e3 − 1.55e3i)15-s − 5.06e3·16-s − 1.20e4i·17-s + ⋯
L(s)  = 1  − 1.57i·2-s − 0.577i·3-s − 1.47·4-s + (0.205 − 0.978i)5-s − 0.907·6-s + 1.35i·7-s + 0.738i·8-s − 0.333·9-s + (−1.53 − 0.323i)10-s + 0.275·11-s + 0.848i·12-s − 1.70i·13-s + 2.13·14-s + (−0.565 − 0.118i)15-s − 0.308·16-s − 0.597i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.205i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.978 - 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $-0.978 - 0.205i$
Analytic conductor: \(4.68577\)
Root analytic conductor: \(2.16466\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :7/2),\ -0.978 - 0.205i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.140204 + 1.34857i\)
\(L(\frac12)\) \(\approx\) \(0.140204 + 1.34857i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 27iT \)
5 \( 1 + (-57.4 + 273. i)T \)
good2 \( 1 + 17.7iT - 128T^{2} \)
7 \( 1 - 1.23e3iT - 8.23e5T^{2} \)
11 \( 1 - 1.21e3T + 1.94e7T^{2} \)
13 \( 1 + 1.35e4iT - 6.27e7T^{2} \)
17 \( 1 + 1.20e4iT - 4.10e8T^{2} \)
19 \( 1 - 3.28e4T + 8.93e8T^{2} \)
23 \( 1 - 9.42e3iT - 3.40e9T^{2} \)
29 \( 1 - 7.99e4T + 1.72e10T^{2} \)
31 \( 1 - 1.13e5T + 2.75e10T^{2} \)
37 \( 1 - 5.18e5iT - 9.49e10T^{2} \)
41 \( 1 - 6.04e5T + 1.94e11T^{2} \)
43 \( 1 + 2.91e5iT - 2.71e11T^{2} \)
47 \( 1 + 1.09e6iT - 5.06e11T^{2} \)
53 \( 1 - 3.05e5iT - 1.17e12T^{2} \)
59 \( 1 + 1.44e5T + 2.48e12T^{2} \)
61 \( 1 + 8.00e4T + 3.14e12T^{2} \)
67 \( 1 - 3.79e6iT - 6.06e12T^{2} \)
71 \( 1 - 3.66e6T + 9.09e12T^{2} \)
73 \( 1 - 1.67e6iT - 1.10e13T^{2} \)
79 \( 1 + 3.76e6T + 1.92e13T^{2} \)
83 \( 1 - 1.77e6iT - 2.71e13T^{2} \)
89 \( 1 + 4.41e6T + 4.42e13T^{2} \)
97 \( 1 - 4.60e6iT - 8.07e13T^{2} \)
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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

−17.67265084260284939308125492369, −15.64917163460973452792046371781, −13.51751364019418356548337161476, −12.45385650518820746335479342197, −11.73783287282642311097357926754, −9.796681432451493697699143613069, −8.482496132848032876617861973803, −5.36399523029994110584349003619, −2.74274877496899097665027724920, −0.953046337948809790252220674156, 4.22220423891170523818833110192, 6.36907659384206059251473587088, 7.48959171795985101714205964839, 9.479392567865659343612725815761, 11.06725465341160243387854588104, 13.98603980755424067437475965143, 14.33838198740892458262410255621, 15.89279851394759522995092248160, 16.83798856339034651587528160613, 17.85095227886717253230004196618

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