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Properties

Label 2-10-5.4-c9-0-0
Degree $2$
Conductor $10$
Sign $-0.231 - 0.972i$
Analytic cond. $5.15035$
Root an. cond. $2.26944$
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  − 16i·2-s + 102. i·3-s − 256·4-s + (−1.35e3 + 324. i)5-s + 1.64e3·6-s + 1.05e4i·7-s + 4.09e3i·8-s + 9.15e3·9-s + (5.18e3 + 2.17e4i)10-s − 7.44e4·11-s − 2.62e4i·12-s − 5.22e4i·13-s + 1.69e5·14-s + (−3.32e4 − 1.39e5i)15-s + 6.55e4·16-s − 7.65e4i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.731i·3-s − 0.5·4-s + (−0.972 + 0.231i)5-s + 0.517·6-s + 1.66i·7-s + 0.353i·8-s + 0.465·9-s + (0.164 + 0.687i)10-s − 1.53·11-s − 0.365i·12-s − 0.507i·13-s + 1.17·14-s + (−0.169 − 0.711i)15-s + 0.250·16-s − 0.222i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.231 - 0.972i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.231 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $-0.231 - 0.972i$
Analytic conductor: \(5.15035\)
Root analytic conductor: \(2.26944\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :9/2),\ -0.231 - 0.972i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.515140 + 0.652431i\)
\(L(\frac12)\) \(\approx\) \(0.515140 + 0.652431i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 16iT \)
5 \( 1 + (1.35e3 - 324. i)T \)
good3 \( 1 - 102. iT - 1.96e4T^{2} \)
7 \( 1 - 1.05e4iT - 4.03e7T^{2} \)
11 \( 1 + 7.44e4T + 2.35e9T^{2} \)
13 \( 1 + 5.22e4iT - 1.06e10T^{2} \)
17 \( 1 + 7.65e4iT - 1.18e11T^{2} \)
19 \( 1 + 1.49e4T + 3.22e11T^{2} \)
23 \( 1 - 1.10e6iT - 1.80e12T^{2} \)
29 \( 1 + 7.50e5T + 1.45e13T^{2} \)
31 \( 1 - 6.88e6T + 2.64e13T^{2} \)
37 \( 1 - 2.04e7iT - 1.29e14T^{2} \)
41 \( 1 + 3.53e6T + 3.27e14T^{2} \)
43 \( 1 - 8.15e6iT - 5.02e14T^{2} \)
47 \( 1 - 2.14e7iT - 1.11e15T^{2} \)
53 \( 1 + 9.48e7iT - 3.29e15T^{2} \)
59 \( 1 - 3.55e7T + 8.66e15T^{2} \)
61 \( 1 + 5.08e7T + 1.16e16T^{2} \)
67 \( 1 - 1.73e7iT - 2.72e16T^{2} \)
71 \( 1 - 2.64e8T + 4.58e16T^{2} \)
73 \( 1 - 3.41e8iT - 5.88e16T^{2} \)
79 \( 1 + 2.08e8T + 1.19e17T^{2} \)
83 \( 1 + 3.81e8iT - 1.86e17T^{2} \)
89 \( 1 + 8.81e8T + 3.50e17T^{2} \)
97 \( 1 - 4.84e8iT - 7.60e17T^{2} \)
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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

−19.04993075810108831946973378366, −18.23954319058705672857125019226, −15.76260028186734263186477982161, −15.24859386750862650696946236340, −12.82335275556508879722855856083, −11.49999631144230547647770843971, −9.993722822478043476379211024327, −8.237634649708821952232549981708, −5.03399840262277744389502857975, −2.93958597831388971195139752913, 0.52097520935218875100403050296, 4.33547937838848091603762914151, 7.06835646614364493304219986721, 7.952071566358300412624674409202, 10.53502779041389204895393382805, 12.68211263888722155807232384299, 13.80038183841276154018164380699, 15.63499536637504678767637221006, 16.76989439181311579120494820610, 18.29054197065291252499256152090

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