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Properties

Label 2-10-5.3-c10-0-4
Degree $2$
Conductor $10$
Sign $-0.916 - 0.400i$
Analytic cond. $6.35357$
Root an. cond. $2.52062$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (16 − 16i)2-s + (−183 − 183i)3-s − 512i·4-s + (−1.87e3 + 2.50e3i)5-s − 5.85e3·6-s + (−8.40e3 + 8.40e3i)7-s + (−8.19e3 − 8.19e3i)8-s + 7.92e3i·9-s + (1.00e4 + 7.00e4i)10-s − 1.73e5·11-s + (−9.36e4 + 9.36e4i)12-s + (−2.32e5 − 2.32e5i)13-s + 2.69e5i·14-s + (8.00e5 − 1.14e5i)15-s − 2.62e5·16-s + (1.88e6 − 1.88e6i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (−0.753 − 0.753i)3-s − 0.5i·4-s + (−0.600 + 0.800i)5-s − 0.753·6-s + (−0.500 + 0.500i)7-s + (−0.250 − 0.250i)8-s + 0.134i·9-s + (0.100 + 0.700i)10-s − 1.07·11-s + (−0.376 + 0.376i)12-s + (−0.626 − 0.626i)13-s + 0.500i·14-s + (1.05 − 0.150i)15-s − 0.250·16-s + (1.32 − 1.32i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.916 - 0.400i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.916 - 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $-0.916 - 0.400i$
Analytic conductor: \(6.35357\)
Root analytic conductor: \(2.52062\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :5),\ -0.916 - 0.400i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.0859747 + 0.411764i\)
\(L(\frac12)\) \(\approx\) \(0.0859747 + 0.411764i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-16 + 16i)T \)
5 \( 1 + (1.87e3 - 2.50e3i)T \)
good3 \( 1 + (183 + 183i)T + 5.90e4iT^{2} \)
7 \( 1 + (8.40e3 - 8.40e3i)T - 2.82e8iT^{2} \)
11 \( 1 + 1.73e5T + 2.59e10T^{2} \)
13 \( 1 + (2.32e5 + 2.32e5i)T + 1.37e11iT^{2} \)
17 \( 1 + (-1.88e6 + 1.88e6i)T - 2.01e12iT^{2} \)
19 \( 1 - 1.10e6iT - 6.13e12T^{2} \)
23 \( 1 + (5.22e6 + 5.22e6i)T + 4.14e13iT^{2} \)
29 \( 1 - 2.47e7iT - 4.20e14T^{2} \)
31 \( 1 + 1.00e7T + 8.19e14T^{2} \)
37 \( 1 + (-5.63e7 + 5.63e7i)T - 4.80e15iT^{2} \)
41 \( 1 + 1.53e8T + 1.34e16T^{2} \)
43 \( 1 + (-5.93e7 - 5.93e7i)T + 2.16e16iT^{2} \)
47 \( 1 + (-1.72e8 + 1.72e8i)T - 5.25e16iT^{2} \)
53 \( 1 + (-1.96e8 - 1.96e8i)T + 1.74e17iT^{2} \)
59 \( 1 + 6.94e8iT - 5.11e17T^{2} \)
61 \( 1 - 9.06e8T + 7.13e17T^{2} \)
67 \( 1 + (9.62e8 - 9.62e8i)T - 1.82e18iT^{2} \)
71 \( 1 + 3.12e9T + 3.25e18T^{2} \)
73 \( 1 + (6.36e8 + 6.36e8i)T + 4.29e18iT^{2} \)
79 \( 1 - 1.96e9iT - 9.46e18T^{2} \)
83 \( 1 + (5.18e9 + 5.18e9i)T + 1.55e19iT^{2} \)
89 \( 1 + 7.77e9iT - 3.11e19T^{2} \)
97 \( 1 + (6.40e8 - 6.40e8i)T - 7.37e19iT^{2} \)
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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

−18.23498356647136324701241640361, −16.06842489748991662543680974712, −14.55785196726634088092852042834, −12.69033673523140136186095112993, −11.83386948381122037982389703625, −10.26104920901986566020196732926, −7.35077607274388347352801449420, −5.63607573809750764452176700342, −2.91353861480390921986173232870, −0.21298830104623525771142697366, 4.11258631128076034722736884796, 5.51890087363622651618771930643, 7.82031297319141331675296873376, 10.08791032879372778294701243055, 11.86043780059560216164986306203, 13.29615959638143144285255479439, 15.31579883164481178863418187891, 16.38129556966235166425389175243, 17.09951454350802975732890847444, 19.35384534432907741148630021168

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