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Properties

Label 2-10-5.3-c4-0-0
Degree $2$
Conductor $10$
Sign $0.640 - 0.767i$
Analytic cond. $1.03369$
Root an. cond. $1.01671$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 + 2i)2-s + (9 + 9i)3-s − 8i·4-s + (−15 − 20i)5-s − 36·6-s + (29 − 29i)7-s + (16 + 16i)8-s + 81i·9-s + (70 + 10i)10-s − 118·11-s + (72 − 72i)12-s + (69 + 69i)13-s + 116i·14-s + (45 − 315i)15-s − 64·16-s + (−271 + 271i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (1 + i)3-s − 0.5i·4-s + (−0.599 − 0.800i)5-s − 6-s + (0.591 − 0.591i)7-s + (0.250 + 0.250i)8-s + i·9-s + (0.700 + 0.100i)10-s − 0.975·11-s + (0.5 − 0.5i)12-s + (0.408 + 0.408i)13-s + 0.591i·14-s + (0.200 − 1.39i)15-s − 0.250·16-s + (−0.937 + 0.937i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.640 - 0.767i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.640 - 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $0.640 - 0.767i$
Analytic conductor: \(1.03369\)
Root analytic conductor: \(1.01671\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :2),\ 0.640 - 0.767i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.905951 + 0.423920i\)
\(L(\frac12)\) \(\approx\) \(0.905951 + 0.423920i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2 - 2i)T \)
5 \( 1 + (15 + 20i)T \)
good3 \( 1 + (-9 - 9i)T + 81iT^{2} \)
7 \( 1 + (-29 + 29i)T - 2.40e3iT^{2} \)
11 \( 1 + 118T + 1.46e4T^{2} \)
13 \( 1 + (-69 - 69i)T + 2.85e4iT^{2} \)
17 \( 1 + (271 - 271i)T - 8.35e4iT^{2} \)
19 \( 1 + 280iT - 1.30e5T^{2} \)
23 \( 1 + (-269 - 269i)T + 2.79e5iT^{2} \)
29 \( 1 + 680iT - 7.07e5T^{2} \)
31 \( 1 - 202T + 9.23e5T^{2} \)
37 \( 1 + (651 - 651i)T - 1.87e6iT^{2} \)
41 \( 1 - 1.68e3T + 2.82e6T^{2} \)
43 \( 1 + (-1.08e3 - 1.08e3i)T + 3.41e6iT^{2} \)
47 \( 1 + (-1.26e3 + 1.26e3i)T - 4.87e6iT^{2} \)
53 \( 1 + (611 + 611i)T + 7.89e6iT^{2} \)
59 \( 1 + 1.16e3iT - 1.21e7T^{2} \)
61 \( 1 + 5.59e3T + 1.38e7T^{2} \)
67 \( 1 + (751 - 751i)T - 2.01e7iT^{2} \)
71 \( 1 - 6.44e3T + 2.54e7T^{2} \)
73 \( 1 + (2.95e3 + 2.95e3i)T + 2.83e7iT^{2} \)
79 \( 1 + 1.05e4iT - 3.89e7T^{2} \)
83 \( 1 + (6.23e3 + 6.23e3i)T + 4.74e7iT^{2} \)
89 \( 1 - 1.44e4iT - 6.27e7T^{2} \)
97 \( 1 + (7.31e3 - 7.31e3i)T - 8.85e7iT^{2} \)
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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

−20.40927505534607014480708864434, −19.37004415030012598391470815480, −17.39657263892531068990599874961, −15.92584702397339250974258949644, −15.14653380220326052994219103775, −13.50900885747239028179362041665, −10.84596160869241973199237549649, −9.104051683033048217187035624946, −7.961397833103900235480438224705, −4.46846646297317084744423291953, 2.63299884198847838915051193532, 7.41236298354186449641074172392, 8.576419223140330273316556010163, 10.92683241127515175220492978638, 12.56296324811360941388841825951, 14.09689758628053902298710611143, 15.58359723051238049094321415300, 18.17528192393846041571962560345, 18.57683806843483705103387378519, 19.83766310813537809024435622667

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