Properties

Label 2-10-5.4-c17-0-5
Degree $2$
Conductor $10$
Sign $0.204 + 0.978i$
Analytic cond. $18.3222$
Root an. cond. $4.28044$
Motivic weight $17$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 256i·2-s − 2.97e3i·3-s − 6.55e4·4-s + (8.55e5 − 1.78e5i)5-s − 7.61e5·6-s + 7.58e6i·7-s + 1.67e7i·8-s + 1.20e8·9-s + (−4.56e7 − 2.18e8i)10-s + 6.04e8·11-s + 1.94e8i·12-s + 1.32e9i·13-s + 1.94e9·14-s + (−5.29e8 − 2.54e9i)15-s + 4.29e9·16-s − 3.25e10i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.261i·3-s − 0.5·4-s + (0.978 − 0.204i)5-s − 0.185·6-s + 0.497i·7-s + 0.353i·8-s + 0.931·9-s + (−0.144 − 0.692i)10-s + 0.850·11-s + 0.130i·12-s + 0.451i·13-s + 0.351·14-s + (−0.0533 − 0.256i)15-s + 0.250·16-s − 1.13i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.204 + 0.978i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $0.204 + 0.978i$
Analytic conductor: \(18.3222\)
Root analytic conductor: \(4.28044\)
Motivic weight: \(17\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :17/2),\ 0.204 + 0.978i)\)

Particular Values

\(L(9)\) \(\approx\) \(1.81225 - 1.47350i\)
\(L(\frac12)\) \(\approx\) \(1.81225 - 1.47350i\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 256iT \)
5 \( 1 + (-8.55e5 + 1.78e5i)T \)
good3 \( 1 + 2.97e3iT - 1.29e8T^{2} \)
7 \( 1 - 7.58e6iT - 2.32e14T^{2} \)
11 \( 1 - 6.04e8T + 5.05e17T^{2} \)
13 \( 1 - 1.32e9iT - 8.65e18T^{2} \)
17 \( 1 + 3.25e10iT - 8.27e20T^{2} \)
19 \( 1 + 1.67e10T + 5.48e21T^{2} \)
23 \( 1 + 7.14e11iT - 1.41e23T^{2} \)
29 \( 1 - 1.25e12T + 7.25e24T^{2} \)
31 \( 1 + 1.26e12T + 2.25e25T^{2} \)
37 \( 1 - 3.37e13iT - 4.56e26T^{2} \)
41 \( 1 - 3.95e13T + 2.61e27T^{2} \)
43 \( 1 - 8.15e13iT - 5.87e27T^{2} \)
47 \( 1 + 1.28e14iT - 2.66e28T^{2} \)
53 \( 1 + 4.11e14iT - 2.05e29T^{2} \)
59 \( 1 - 1.32e15T + 1.27e30T^{2} \)
61 \( 1 + 1.20e15T + 2.24e30T^{2} \)
67 \( 1 - 2.90e15iT - 1.10e31T^{2} \)
71 \( 1 + 7.20e15T + 2.96e31T^{2} \)
73 \( 1 + 4.16e15iT - 4.74e31T^{2} \)
79 \( 1 + 4.74e15T + 1.81e32T^{2} \)
83 \( 1 + 1.57e16iT - 4.21e32T^{2} \)
89 \( 1 + 2.48e16T + 1.37e33T^{2} \)
97 \( 1 - 1.31e17iT - 5.95e33T^{2} \)
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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

−16.48933444199485118850181444370, −14.44341584398359964837661138373, −13.14728236682331174017538553687, −11.93711249340793870061661592342, −10.10435441904181520937616774475, −8.913881678833064719811022772368, −6.55019203329067982346372119409, −4.62655140148401512393662745612, −2.40904194911407649297790750128, −1.08256255985371387724378603358, 1.40450319184057051952106837257, 3.95586427532170979313041539800, 5.79486106919323572104792799010, 7.25376612385907582660267082199, 9.267521848707955243908360467334, 10.49056199909863587028191609145, 12.88685146453811381280499435816, 14.12747996649013267168392202403, 15.44315370536183816961476113892, 16.95707328425475168837247260467

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