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Properties

Label 2-15-15.2-c9-0-5
Degree $2$
Conductor $15$
Sign $-0.355 - 0.934i$
Analytic cond. $7.72553$
Root an. cond. $2.77948$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−10.0 + 10.0i)2-s + (112. + 84.4i)3-s + 311. i·4-s + (1.25e3 − 615. i)5-s + (−1.96e3 + 277. i)6-s + (2.70e3 + 2.70e3i)7-s + (−8.24e3 − 8.24e3i)8-s + (5.43e3 + 1.89e4i)9-s + (−6.41e3 + 1.87e4i)10-s + 2.51e4i·11-s + (−2.62e4 + 3.48e4i)12-s + (−1.31e5 + 1.31e5i)13-s − 5.42e4·14-s + (1.92e5 + 3.69e4i)15-s + 6.08e3·16-s + (2.74e5 − 2.74e5i)17-s + ⋯
L(s)  = 1  + (−0.442 + 0.442i)2-s + (0.798 + 0.601i)3-s + 0.607i·4-s + (0.897 − 0.440i)5-s + (−0.620 + 0.0872i)6-s + (0.426 + 0.426i)7-s + (−0.712 − 0.712i)8-s + (0.275 + 0.961i)9-s + (−0.202 + 0.592i)10-s + 0.518i·11-s + (−0.365 + 0.485i)12-s + (−1.28 + 1.28i)13-s − 0.377·14-s + (0.982 + 0.188i)15-s + 0.0231·16-s + (0.796 − 0.796i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.355 - 0.934i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.355 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $-0.355 - 0.934i$
Analytic conductor: \(7.72553\)
Root analytic conductor: \(2.77948\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :9/2),\ -0.355 - 0.934i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.05528 + 1.53092i\)
\(L(\frac12)\) \(\approx\) \(1.05528 + 1.53092i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-112. - 84.4i)T \)
5 \( 1 + (-1.25e3 + 615. i)T \)
good2 \( 1 + (10.0 - 10.0i)T - 512iT^{2} \)
7 \( 1 + (-2.70e3 - 2.70e3i)T + 4.03e7iT^{2} \)
11 \( 1 - 2.51e4iT - 2.35e9T^{2} \)
13 \( 1 + (1.31e5 - 1.31e5i)T - 1.06e10iT^{2} \)
17 \( 1 + (-2.74e5 + 2.74e5i)T - 1.18e11iT^{2} \)
19 \( 1 + 5.04e5iT - 3.22e11T^{2} \)
23 \( 1 + (-3.57e5 - 3.57e5i)T + 1.80e12iT^{2} \)
29 \( 1 - 1.44e6T + 1.45e13T^{2} \)
31 \( 1 + 1.57e6T + 2.64e13T^{2} \)
37 \( 1 + (-1.41e7 - 1.41e7i)T + 1.29e14iT^{2} \)
41 \( 1 + 1.52e7iT - 3.27e14T^{2} \)
43 \( 1 + (-1.45e7 + 1.45e7i)T - 5.02e14iT^{2} \)
47 \( 1 + (-2.49e7 + 2.49e7i)T - 1.11e15iT^{2} \)
53 \( 1 + (-2.38e7 - 2.38e7i)T + 3.29e15iT^{2} \)
59 \( 1 + 1.41e8T + 8.66e15T^{2} \)
61 \( 1 - 2.77e7T + 1.16e16T^{2} \)
67 \( 1 + (-5.16e7 - 5.16e7i)T + 2.72e16iT^{2} \)
71 \( 1 + 2.50e8iT - 4.58e16T^{2} \)
73 \( 1 + (8.81e7 - 8.81e7i)T - 5.88e16iT^{2} \)
79 \( 1 + 3.89e8iT - 1.19e17T^{2} \)
83 \( 1 + (-9.47e6 - 9.47e6i)T + 1.86e17iT^{2} \)
89 \( 1 - 4.88e8T + 3.50e17T^{2} \)
97 \( 1 + (-7.00e7 - 7.00e7i)T + 7.60e17iT^{2} \)
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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

−17.31463529022293227664422071321, −16.41656201013714454519676009225, −14.99605401533011419959994979924, −13.70129594153167817952541498790, −12.10247585385472677440245798126, −9.690075906518615813096773052789, −8.926430635303125133043955648007, −7.30470235735191482854362928456, −4.77722911082394431080073010927, −2.43214907663014399601311997310, 1.13310764978219962741165851213, 2.67161295645059634535694866484, 5.87143290832810488393358891905, 7.85898293120678670045313790490, 9.589615388101966681450936804583, 10.63865632745722630759435410261, 12.65816826184688761583398150795, 14.25864835991264477833488871552, 14.79593978726021939596091284971, 17.31851840762455552820315561485

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