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Properties

Label 2-10-5.4-c15-0-2
Degree $2$
Conductor $10$
Sign $0.392 - 0.919i$
Analytic cond. $14.2693$
Root an. cond. $3.77747$
Motivic weight $15$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 128i·2-s − 2.93e3i·3-s − 1.63e4·4-s + (−1.60e5 − 6.84e4i)5-s + 3.75e5·6-s + 1.80e6i·7-s − 2.09e6i·8-s + 5.75e6·9-s + (8.76e6 − 2.05e7i)10-s + 8.15e7·11-s + 4.80e7i·12-s + 1.37e8i·13-s − 2.30e8·14-s + (−2.00e8 + 4.71e8i)15-s + 2.68e8·16-s + 2.67e9i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.774i·3-s − 0.5·4-s + (−0.919 − 0.392i)5-s + 0.547·6-s + 0.826i·7-s − 0.353i·8-s + 0.400·9-s + (0.277 − 0.650i)10-s + 1.26·11-s + 0.387i·12-s + 0.605i·13-s − 0.584·14-s + (−0.303 + 0.712i)15-s + 0.250·16-s + 1.58i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.392 - 0.919i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (0.392 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $0.392 - 0.919i$
Analytic conductor: \(14.2693\)
Root analytic conductor: \(3.77747\)
Motivic weight: \(15\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :15/2),\ 0.392 - 0.919i)\)

Particular Values

\(L(8)\) \(\approx\) \(1.23140 + 0.813725i\)
\(L(\frac12)\) \(\approx\) \(1.23140 + 0.813725i\)
\(L(\frac{17}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 128iT \)
5 \( 1 + (1.60e5 + 6.84e4i)T \)
good3 \( 1 + 2.93e3iT - 1.43e7T^{2} \)
7 \( 1 - 1.80e6iT - 4.74e12T^{2} \)
11 \( 1 - 8.15e7T + 4.17e15T^{2} \)
13 \( 1 - 1.37e8iT - 5.11e16T^{2} \)
17 \( 1 - 2.67e9iT - 2.86e18T^{2} \)
19 \( 1 - 2.47e9T + 1.51e19T^{2} \)
23 \( 1 - 2.86e9iT - 2.66e20T^{2} \)
29 \( 1 + 1.20e11T + 8.62e21T^{2} \)
31 \( 1 - 1.95e11T + 2.34e22T^{2} \)
37 \( 1 + 2.27e11iT - 3.33e23T^{2} \)
41 \( 1 - 1.53e12T + 1.55e24T^{2} \)
43 \( 1 - 2.06e12iT - 3.17e24T^{2} \)
47 \( 1 - 2.80e12iT - 1.20e25T^{2} \)
53 \( 1 + 5.32e12iT - 7.31e25T^{2} \)
59 \( 1 + 1.26e13T + 3.65e26T^{2} \)
61 \( 1 + 3.42e13T + 6.02e26T^{2} \)
67 \( 1 - 5.27e13iT - 2.46e27T^{2} \)
71 \( 1 - 2.86e13T + 5.87e27T^{2} \)
73 \( 1 - 1.30e14iT - 8.90e27T^{2} \)
79 \( 1 + 1.18e14T + 2.91e28T^{2} \)
83 \( 1 + 4.70e14iT - 6.11e28T^{2} \)
89 \( 1 - 5.97e14T + 1.74e29T^{2} \)
97 \( 1 - 9.88e14iT - 6.33e29T^{2} \)
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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

−17.25554082294801727266517299776, −15.87334770776817871449344522479, −14.63711387076268010493899232265, −12.85795930391923849458733963015, −11.78375602974404385720227630205, −9.095686442129767313998788099008, −7.71899977220755141024084619124, −6.25751981155143702998880981447, −4.12878123722802406510614455531, −1.32059913438675733090261316158, 0.74201540605701600353620405410, 3.38854029011325120486098495479, 4.48958801595786704545969091624, 7.32798930922584770233204787129, 9.429881813399963121325632179836, 10.75290755944848218091269569943, 11.96713607389964167204245514023, 13.90187474797016841518242292694, 15.36067280563943669198091447655, 16.72372600272119938795179406531

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