Properties

Label 2-10-5.2-c10-0-1
Degree $2$
Conductor $10$
Sign $0.513 - 0.858i$
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 + (4.29 − 4.29i)3-s + 512i·4-s + (−2.97e3 − 946. i)5-s − 137.·6-s + (2.12e4 + 2.12e4i)7-s + (8.19e3 − 8.19e3i)8-s + 5.90e4i·9-s + (3.25e4 + 6.27e4i)10-s + 1.55e5·11-s + (2.19e3 + 2.19e3i)12-s + (−3.58e5 + 3.58e5i)13-s − 6.81e5i·14-s + (−1.68e4 + 8.72e3i)15-s − 2.62e5·16-s + (−6.09e5 − 6.09e5i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (0.0176 − 0.0176i)3-s + 0.5i·4-s + (−0.953 − 0.302i)5-s − 0.0176·6-s + (1.26 + 1.26i)7-s + (0.250 − 0.250i)8-s + 0.999i·9-s + (0.325 + 0.627i)10-s + 0.966·11-s + (0.00883 + 0.00883i)12-s + (−0.965 + 0.965i)13-s − 1.26i·14-s + (−0.0221 + 0.0114i)15-s − 0.250·16-s + (−0.429 − 0.429i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 - 0.858i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.513 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $0.513 - 0.858i$
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: \((2,\ 10,\ (\ :5),\ 0.513 - 0.858i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.868782 + 0.492475i\)
\(L(\frac12)\) \(\approx\) \(0.868782 + 0.492475i\)
\(L(6)\) 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 \( 1 + (2.97e3 + 946. i)T \)
good3 \( 1 + (-4.29 + 4.29i)T - 5.90e4iT^{2} \)
7 \( 1 + (-2.12e4 - 2.12e4i)T + 2.82e8iT^{2} \)
11 \( 1 - 1.55e5T + 2.59e10T^{2} \)
13 \( 1 + (3.58e5 - 3.58e5i)T - 1.37e11iT^{2} \)
17 \( 1 + (6.09e5 + 6.09e5i)T + 2.01e12iT^{2} \)
19 \( 1 - 3.35e5iT - 6.13e12T^{2} \)
23 \( 1 + (5.50e6 - 5.50e6i)T - 4.14e13iT^{2} \)
29 \( 1 - 1.33e6iT - 4.20e14T^{2} \)
31 \( 1 - 2.59e7T + 8.19e14T^{2} \)
37 \( 1 + (5.50e7 + 5.50e7i)T + 4.80e15iT^{2} \)
41 \( 1 - 1.37e8T + 1.34e16T^{2} \)
43 \( 1 + (9.34e7 - 9.34e7i)T - 2.16e16iT^{2} \)
47 \( 1 + (-1.14e8 - 1.14e8i)T + 5.25e16iT^{2} \)
53 \( 1 + (2.29e7 - 2.29e7i)T - 1.74e17iT^{2} \)
59 \( 1 + 8.73e8iT - 5.11e17T^{2} \)
61 \( 1 - 5.87e8T + 7.13e17T^{2} \)
67 \( 1 + (6.75e8 + 6.75e8i)T + 1.82e18iT^{2} \)
71 \( 1 + 5.54e8T + 3.25e18T^{2} \)
73 \( 1 + (-8.91e8 + 8.91e8i)T - 4.29e18iT^{2} \)
79 \( 1 + 1.69e9iT - 9.46e18T^{2} \)
83 \( 1 + (-1.96e9 + 1.96e9i)T - 1.55e19iT^{2} \)
89 \( 1 + 7.73e9iT - 3.11e19T^{2} \)
97 \( 1 + (-1.16e10 - 1.16e10i)T + 7.37e19iT^{2} \)
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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

−18.91890306606713708585990862336, −17.42270554984396288850162042356, −15.94998245901884839040067187990, −14.37413942709943842077463864238, −12.06699963167518199879033590377, −11.38196318803325814749601785513, −9.050330583660059136178110773157, −7.76571984768246298819813041877, −4.66795959299419327147647536599, −1.95920047526599677587289107791, 0.70157569794573947950110322850, 4.23286549249332268627312822473, 6.96625061099426196784129441112, 8.266463606202724761740674957189, 10.40186380542329344386760003187, 11.88153458433276470496808570065, 14.34064660771885373094316011046, 15.20263164078837613190088056172, 17.03120847333094738199212045101, 17.89374135148531759578659611216

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