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Properties

Label 2-15-15.14-c6-0-3
Degree $2$
Conductor $15$
Sign $0.122 - 0.992i$
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  − 2.12·2-s + (19.8 + 18.2i)3-s − 59.4·4-s + (72.7 + 101. i)5-s + (−42.2 − 38.8i)6-s + 435. i·7-s + 262.·8-s + (60.6 + 726. i)9-s + (−154. − 216. i)10-s − 1.65e3i·11-s + (−1.18e3 − 1.08e3i)12-s − 1.64e3i·13-s − 926. i·14-s + (−412. + 3.34e3i)15-s + 3.24e3·16-s + 2.05e3·17-s + ⋯
L(s)  = 1  − 0.265·2-s + (0.735 + 0.677i)3-s − 0.929·4-s + (0.582 + 0.813i)5-s + (−0.195 − 0.180i)6-s + 1.26i·7-s + 0.512·8-s + (0.0831 + 0.996i)9-s + (−0.154 − 0.216i)10-s − 1.24i·11-s + (−0.683 − 0.629i)12-s − 0.747i·13-s − 0.337i·14-s + (−0.122 + 0.992i)15-s + 0.792·16-s + 0.417·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.122 - 0.992i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.122 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.122 - 0.992i$
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: \((2,\ 15,\ (\ :3),\ 0.122 - 0.992i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.01835 + 0.900731i\)
\(L(\frac12)\) \(\approx\) \(1.01835 + 0.900731i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-19.8 - 18.2i)T \)
5 \( 1 + (-72.7 - 101. i)T \)
good2 \( 1 + 2.12T + 64T^{2} \)
7 \( 1 - 435. iT - 1.17e5T^{2} \)
11 \( 1 + 1.65e3iT - 1.77e6T^{2} \)
13 \( 1 + 1.64e3iT - 4.82e6T^{2} \)
17 \( 1 - 2.05e3T + 2.41e7T^{2} \)
19 \( 1 + 3.63e3T + 4.70e7T^{2} \)
23 \( 1 - 2.12e4T + 1.48e8T^{2} \)
29 \( 1 + 1.14e4iT - 5.94e8T^{2} \)
31 \( 1 - 1.34e4T + 8.87e8T^{2} \)
37 \( 1 + 1.77e3iT - 2.56e9T^{2} \)
41 \( 1 + 2.29e3iT - 4.75e9T^{2} \)
43 \( 1 + 1.44e4iT - 6.32e9T^{2} \)
47 \( 1 - 4.96e4T + 1.07e10T^{2} \)
53 \( 1 + 8.17e4T + 2.21e10T^{2} \)
59 \( 1 + 2.00e5iT - 4.21e10T^{2} \)
61 \( 1 - 6.37e4T + 5.15e10T^{2} \)
67 \( 1 + 3.51e5iT - 9.04e10T^{2} \)
71 \( 1 + 4.04e5iT - 1.28e11T^{2} \)
73 \( 1 - 6.66e5iT - 1.51e11T^{2} \)
79 \( 1 + 7.58e5T + 2.43e11T^{2} \)
83 \( 1 + 2.61e5T + 3.26e11T^{2} \)
89 \( 1 - 3.16e5iT - 4.96e11T^{2} \)
97 \( 1 - 3.20e5iT - 8.32e11T^{2} \)
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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

−18.65218976809502520016480329012, −17.12875488285607644861456543179, −15.38807936845535714034182108250, −14.35571186839493425227026241647, −13.17182001801708169290434611204, −10.77807238840336208604568015568, −9.406251397713295330337198839099, −8.348258755528911077342083124263, −5.48925040432443040983883793623, −3.03713602588289620115049922738, 1.20915478528197878840399843299, 4.43772078034011480427173897634, 7.23561812431648800926815858812, 8.832030858215381866728900644541, 9.960874403842509832294131211674, 12.65719457887522809582817055628, 13.49956024450133382599484628120, 14.55979658503214938565620320320, 16.91670416815337394729792062940, 17.63740556196945535160301042184

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