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Properties

Label 2-10-5.4-c15-0-0
Degree $2$
Conductor $10$
Sign $-0.370 - 0.928i$
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 + 5.60e3i·3-s − 1.63e4·4-s + (1.62e5 − 6.46e4i)5-s + 7.17e5·6-s + 7.50e5i·7-s + 2.09e6i·8-s − 1.70e7·9-s + (−8.27e6 − 2.07e7i)10-s − 6.42e7·11-s − 9.18e7i·12-s + 3.28e8i·13-s + 9.61e7·14-s + (3.62e8 + 9.09e8i)15-s + 2.68e8·16-s + 6.22e8i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.47i·3-s − 0.5·4-s + (0.928 − 0.370i)5-s + 1.04·6-s + 0.344i·7-s + 0.353i·8-s − 1.18·9-s + (−0.261 − 0.656i)10-s − 0.994·11-s − 0.739i·12-s + 1.45i·13-s + 0.243·14-s + (0.547 + 1.37i)15-s + 0.250·16-s + 0.367i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.370 - 0.928i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.370 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $-0.370 - 0.928i$
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: \((2,\ 10,\ (\ :15/2),\ -0.370 - 0.928i)\)

Particular Values

\(L(8)\) \(\approx\) \(0.763570 + 1.12619i\)
\(L(\frac12)\) \(\approx\) \(0.763570 + 1.12619i\)
\(L(\frac{17}{2})\) 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 \( 1 + (-1.62e5 + 6.46e4i)T \)
good3 \( 1 - 5.60e3iT - 1.43e7T^{2} \)
7 \( 1 - 7.50e5iT - 4.74e12T^{2} \)
11 \( 1 + 6.42e7T + 4.17e15T^{2} \)
13 \( 1 - 3.28e8iT - 5.11e16T^{2} \)
17 \( 1 - 6.22e8iT - 2.86e18T^{2} \)
19 \( 1 + 4.73e9T + 1.51e19T^{2} \)
23 \( 1 - 1.90e10iT - 2.66e20T^{2} \)
29 \( 1 + 1.83e11T + 8.62e21T^{2} \)
31 \( 1 - 1.39e11T + 2.34e22T^{2} \)
37 \( 1 + 5.52e11iT - 3.33e23T^{2} \)
41 \( 1 - 6.85e11T + 1.55e24T^{2} \)
43 \( 1 - 1.71e11iT - 3.17e24T^{2} \)
47 \( 1 - 5.47e12iT - 1.20e25T^{2} \)
53 \( 1 + 8.40e12iT - 7.31e25T^{2} \)
59 \( 1 - 2.89e13T + 3.65e26T^{2} \)
61 \( 1 - 4.32e13T + 6.02e26T^{2} \)
67 \( 1 - 3.51e13iT - 2.46e27T^{2} \)
71 \( 1 + 5.12e13T + 5.87e27T^{2} \)
73 \( 1 - 1.73e13iT - 8.90e27T^{2} \)
79 \( 1 - 1.09e14T + 2.91e28T^{2} \)
83 \( 1 + 1.44e14iT - 6.11e28T^{2} \)
89 \( 1 + 1.30e14T + 1.74e29T^{2} \)
97 \( 1 + 6.80e14iT - 6.33e29T^{2} \)
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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

−17.37919405883836466251869933213, −16.14468878718491475736849807010, −14.63808258182732455748764729990, −13.07418136166435879046967856248, −11.13724115183203625635722738889, −9.885835669841670149844859567983, −8.943356351757036166591661910343, −5.50441415941067773667766213239, −4.12940884880639566527004638169, −2.14981729922363673213512751044, 0.53737287743902283786433586192, 2.41089025926777390271600539112, 5.63175135186130461189813738602, 6.92340125816288724061045529510, 8.186213495382472500627163540356, 10.40942160206515201075542658032, 12.83841366464138504149317394823, 13.47209047287318071285702567739, 14.96891184253537766678790793072, 17.03019804699015107741069240979

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