Properties

Label 2-15-5.2-c10-0-3
Degree $2$
Conductor $15$
Sign $-0.515 + 0.856i$
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  + (−41.2 − 41.2i)2-s + (−99.2 + 99.2i)3-s + 2.37e3i·4-s + (953. − 2.97e3i)5-s + 8.17e3·6-s + (1.67e4 + 1.67e4i)7-s + (5.56e4 − 5.56e4i)8-s − 1.96e4i·9-s + (−1.62e5 + 8.33e4i)10-s − 1.43e5·11-s + (−2.35e5 − 2.35e5i)12-s + (4.40e4 − 4.40e4i)13-s − 1.38e6i·14-s + (2.00e5 + 3.89e5i)15-s − 2.16e6·16-s + (5.34e5 + 5.34e5i)17-s + ⋯
L(s)  = 1  + (−1.28 − 1.28i)2-s + (−0.408 + 0.408i)3-s + 2.31i·4-s + (0.305 − 0.952i)5-s + 1.05·6-s + (0.999 + 0.999i)7-s + (1.69 − 1.69i)8-s − 0.333i·9-s + (−1.62 + 0.833i)10-s − 0.888·11-s + (−0.946 − 0.946i)12-s + (0.118 − 0.118i)13-s − 2.57i·14-s + (0.264 + 0.513i)15-s − 2.05·16-s + (0.376 + 0.376i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.515 + 0.856i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.515 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $-0.515 + 0.856i$
Analytic conductor: \(9.53035\)
Root analytic conductor: \(3.08712\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :5),\ -0.515 + 0.856i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.365347 - 0.646495i\)
\(L(\frac12)\) \(\approx\) \(0.365347 - 0.646495i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (99.2 - 99.2i)T \)
5 \( 1 + (-953. + 2.97e3i)T \)
good2 \( 1 + (41.2 + 41.2i)T + 1.02e3iT^{2} \)
7 \( 1 + (-1.67e4 - 1.67e4i)T + 2.82e8iT^{2} \)
11 \( 1 + 1.43e5T + 2.59e10T^{2} \)
13 \( 1 + (-4.40e4 + 4.40e4i)T - 1.37e11iT^{2} \)
17 \( 1 + (-5.34e5 - 5.34e5i)T + 2.01e12iT^{2} \)
19 \( 1 + 4.33e6iT - 6.13e12T^{2} \)
23 \( 1 + (-6.55e6 + 6.55e6i)T - 4.14e13iT^{2} \)
29 \( 1 + 1.25e7iT - 4.20e14T^{2} \)
31 \( 1 - 1.29e7T + 8.19e14T^{2} \)
37 \( 1 + (-1.59e7 - 1.59e7i)T + 4.80e15iT^{2} \)
41 \( 1 + 9.99e7T + 1.34e16T^{2} \)
43 \( 1 + (-1.60e8 + 1.60e8i)T - 2.16e16iT^{2} \)
47 \( 1 + (-5.89e7 - 5.89e7i)T + 5.25e16iT^{2} \)
53 \( 1 + (5.68e7 - 5.68e7i)T - 1.74e17iT^{2} \)
59 \( 1 + 1.00e9iT - 5.11e17T^{2} \)
61 \( 1 - 1.16e9T + 7.13e17T^{2} \)
67 \( 1 + (7.68e8 + 7.68e8i)T + 1.82e18iT^{2} \)
71 \( 1 - 2.18e8T + 3.25e18T^{2} \)
73 \( 1 + (-9.64e8 + 9.64e8i)T - 4.29e18iT^{2} \)
79 \( 1 - 9.54e8iT - 9.46e18T^{2} \)
83 \( 1 + (-2.78e9 + 2.78e9i)T - 1.55e19iT^{2} \)
89 \( 1 + 1.25e9iT - 3.11e19T^{2} \)
97 \( 1 + (-1.56e8 - 1.56e8i)T + 7.37e19iT^{2} \)
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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

−17.07635512273551485692951843372, −15.57453175568139927993957705704, −12.93570717266394619398940317236, −11.79784325939827466998391912389, −10.67477034390390653793821682939, −9.189712693750788080652180218577, −8.264269843217825463818120416194, −4.95597181498691678063122375619, −2.36371678206657007747282229259, −0.66359228748720659304196017693, 1.29425089659320109798732235544, 5.56767539553239993860975056730, 7.13109380990774338492752203450, 7.936549230143711112949385846367, 10.06656667301755580806888409483, 11.01897427536586224891984597985, 13.86715805674149431779613379004, 14.86800835863360061371963449510, 16.39921321148680614139615491283, 17.47711429943932754658364181784

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