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Properties

Label 2-10-5.2-c10-0-4
Degree $2$
Conductor $10$
Sign $-0.235 + 0.971i$
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 + (326. − 326. i)3-s + 512i·4-s + (3.12e3 + 19.4i)5-s − 1.04e4·6-s + (−1.50e3 − 1.50e3i)7-s + (8.19e3 − 8.19e3i)8-s − 1.53e5i·9-s + (−4.96e4 − 5.03e4i)10-s − 7.14e4·11-s + (1.66e5 + 1.66e5i)12-s + (−3.00e5 + 3.00e5i)13-s + 4.82e4i·14-s + (1.02e6 − 1.01e6i)15-s − 2.62e5·16-s + (1.11e6 + 1.11e6i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (1.34 − 1.34i)3-s + 0.5i·4-s + (0.999 + 0.00622i)5-s − 1.34·6-s + (−0.0896 − 0.0896i)7-s + (0.250 − 0.250i)8-s − 2.60i·9-s + (−0.496 − 0.503i)10-s − 0.443·11-s + (0.671 + 0.671i)12-s + (−0.810 + 0.810i)13-s + 0.0896i·14-s + (1.35 − 1.33i)15-s − 0.250·16-s + (0.786 + 0.786i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.235 + 0.971i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.235 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $-0.235 + 0.971i$
Analytic conductor: \(6.35357\)
Root analytic conductor: \(2.52062\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :5),\ -0.235 + 0.971i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.26708 - 1.61131i\)
\(L(\frac12)\) \(\approx\) \(1.26708 - 1.61131i\)
\(L(6)\) 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 \( 1 + (-3.12e3 - 19.4i)T \)
good3 \( 1 + (-326. + 326. i)T - 5.90e4iT^{2} \)
7 \( 1 + (1.50e3 + 1.50e3i)T + 2.82e8iT^{2} \)
11 \( 1 + 7.14e4T + 2.59e10T^{2} \)
13 \( 1 + (3.00e5 - 3.00e5i)T - 1.37e11iT^{2} \)
17 \( 1 + (-1.11e6 - 1.11e6i)T + 2.01e12iT^{2} \)
19 \( 1 + 6.84e5iT - 6.13e12T^{2} \)
23 \( 1 + (-1.52e6 + 1.52e6i)T - 4.14e13iT^{2} \)
29 \( 1 - 2.47e7iT - 4.20e14T^{2} \)
31 \( 1 - 4.13e7T + 8.19e14T^{2} \)
37 \( 1 + (-2.53e7 - 2.53e7i)T + 4.80e15iT^{2} \)
41 \( 1 + 1.32e8T + 1.34e16T^{2} \)
43 \( 1 + (2.46e7 - 2.46e7i)T - 2.16e16iT^{2} \)
47 \( 1 + (-5.19e7 - 5.19e7i)T + 5.25e16iT^{2} \)
53 \( 1 + (3.03e8 - 3.03e8i)T - 1.74e17iT^{2} \)
59 \( 1 - 6.92e7iT - 5.11e17T^{2} \)
61 \( 1 - 6.38e8T + 7.13e17T^{2} \)
67 \( 1 + (-7.98e8 - 7.98e8i)T + 1.82e18iT^{2} \)
71 \( 1 + 4.06e8T + 3.25e18T^{2} \)
73 \( 1 + (5.91e8 - 5.91e8i)T - 4.29e18iT^{2} \)
79 \( 1 + 6.97e8iT - 9.46e18T^{2} \)
83 \( 1 + (2.76e9 - 2.76e9i)T - 1.55e19iT^{2} \)
89 \( 1 + 3.28e9iT - 3.11e19T^{2} \)
97 \( 1 + (3.71e9 + 3.71e9i)T + 7.37e19iT^{2} \)
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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

−18.42737129837583020962953988067, −17.19343219018573176941894432957, −14.58455278271094124838329944050, −13.45244383543495165551042172906, −12.35611035878837756104519566392, −9.807355161279537174493956339026, −8.452448954667566503169682902532, −6.87087864509810877173951289852, −2.76067015636073612727335851885, −1.44638601905441035155284825520, 2.67908604221162747913926324764, 5.15307757553946872242606736930, 8.008669053882301374415523855542, 9.541993365971882840179845743944, 10.22691442270100037944057563854, 13.61159835399907342236239238740, 14.71856597980332698455951825139, 15.82184334025085990364563803225, 17.16825698368099338910427030802, 18.95647852427469805339329692497

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