Properties

Label 2-10-5.2-c18-0-1
Degree $2$
Conductor $10$
Sign $-0.962 - 0.270i$
Analytic cond. $20.5386$
Root an. cond. $4.53195$
Motivic weight $18$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (256 + 256i)2-s + (−5.08e3 + 5.08e3i)3-s + 1.31e5i·4-s + (8.24e4 − 1.95e6i)5-s − 2.60e6·6-s + (2.79e7 + 2.79e7i)7-s + (−3.35e7 + 3.35e7i)8-s + 3.35e8i·9-s + (5.20e8 − 4.78e8i)10-s − 1.32e9·11-s + (−6.65e8 − 6.65e8i)12-s + (−8.52e9 + 8.52e9i)13-s + 1.43e10i·14-s + (9.49e9 + 1.03e10i)15-s − 1.71e10·16-s + (−6.13e10 − 6.13e10i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (−0.258 + 0.258i)3-s + 0.5i·4-s + (0.0422 − 0.999i)5-s − 0.258·6-s + (0.692 + 0.692i)7-s + (−0.250 + 0.250i)8-s + 0.866i·9-s + (0.520 − 0.478i)10-s − 0.563·11-s + (−0.129 − 0.129i)12-s + (−0.804 + 0.804i)13-s + 0.692i·14-s + (0.246 + 0.268i)15-s − 0.250·16-s + (−0.517 − 0.517i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 - 0.270i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (-0.962 - 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $-0.962 - 0.270i$
Analytic conductor: \(20.5386\)
Root analytic conductor: \(4.53195\)
Motivic weight: \(18\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :9),\ -0.962 - 0.270i)\)

Particular Values

\(L(\frac{19}{2})\) \(\approx\) \(0.179910 + 1.30474i\)
\(L(\frac12)\) \(\approx\) \(0.179910 + 1.30474i\)
\(L(10)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-256 - 256i)T \)
5 \( 1 + (-8.24e4 + 1.95e6i)T \)
good3 \( 1 + (5.08e3 - 5.08e3i)T - 3.87e8iT^{2} \)
7 \( 1 + (-2.79e7 - 2.79e7i)T + 1.62e15iT^{2} \)
11 \( 1 + 1.32e9T + 5.55e18T^{2} \)
13 \( 1 + (8.52e9 - 8.52e9i)T - 1.12e20iT^{2} \)
17 \( 1 + (6.13e10 + 6.13e10i)T + 1.40e22iT^{2} \)
19 \( 1 + 1.33e11iT - 1.04e23T^{2} \)
23 \( 1 + (1.46e12 - 1.46e12i)T - 3.24e24iT^{2} \)
29 \( 1 - 2.55e13iT - 2.10e26T^{2} \)
31 \( 1 + 3.39e13T + 6.99e26T^{2} \)
37 \( 1 + (-4.74e13 - 4.74e13i)T + 1.68e28iT^{2} \)
41 \( 1 - 6.29e14T + 1.07e29T^{2} \)
43 \( 1 + (1.52e14 - 1.52e14i)T - 2.52e29iT^{2} \)
47 \( 1 + (-6.93e14 - 6.93e14i)T + 1.25e30iT^{2} \)
53 \( 1 + (8.87e14 - 8.87e14i)T - 1.08e31iT^{2} \)
59 \( 1 + 3.62e15iT - 7.50e31T^{2} \)
61 \( 1 + 1.96e15T + 1.36e32T^{2} \)
67 \( 1 + (3.41e15 + 3.41e15i)T + 7.40e32iT^{2} \)
71 \( 1 - 5.35e16T + 2.10e33T^{2} \)
73 \( 1 + (-3.88e16 + 3.88e16i)T - 3.46e33iT^{2} \)
79 \( 1 + 5.48e16iT - 1.43e34T^{2} \)
83 \( 1 + (2.09e17 - 2.09e17i)T - 3.49e34iT^{2} \)
89 \( 1 + 2.46e17iT - 1.22e35T^{2} \)
97 \( 1 + (-6.86e17 - 6.86e17i)T + 5.77e35iT^{2} \)
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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

−16.64948540062371195521352972334, −15.73037123867450089946339742025, −14.13445997299117629920792878104, −12.70834186900333395722609981057, −11.34405922129634551022180223752, −9.141503018220551374287462375518, −7.67089560468463747062294612065, −5.41671548408491055414554157885, −4.62835602808679222465847047143, −2.07993337623984789980761531377, 0.38765080535758366887551286819, 2.30658236838302347484548795876, 3.99504716755272386750120416555, 5.96732542351304164459381575442, 7.56521625140790610675549133086, 10.10044200727792253459771149344, 11.20466311085268374864805526152, 12.65957485480261301069281320505, 14.22816030423118778896167998712, 15.21455272657297692183888088260

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