↑

Properties

Label 2-10-5.2-c18-0-4
Degree $2$
Conductor $10$
Sign $0.633 - 0.773i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (256 + 256i)2-s + (1.07e4 − 1.07e4i)3-s + 1.31e5i·4-s + (1.75e6 + 8.57e5i)5-s + 5.50e6·6-s + (−4.02e6 − 4.02e6i)7-s + (−3.35e7 + 3.35e7i)8-s + 1.56e8i·9-s + (2.29e8 + 6.68e8i)10-s + 1.03e9·11-s + (1.40e9 + 1.40e9i)12-s + (6.67e9 − 6.67e9i)13-s − 2.06e9i·14-s + (2.80e10 − 9.64e9i)15-s − 1.71e10·16-s + (7.11e10 + 7.11e10i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (0.546 − 0.546i)3-s + 0.5i·4-s + (0.898 + 0.439i)5-s + 0.546·6-s + (−0.0997 − 0.0997i)7-s + (−0.250 + 0.250i)8-s + 0.403i·9-s + (0.229 + 0.668i)10-s + 0.440·11-s + (0.273 + 0.273i)12-s + (0.629 − 0.629i)13-s − 0.0997i·14-s + (0.730 − 0.250i)15-s − 0.250·16-s + (0.600 + 0.600i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.633 - 0.773i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (0.633 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $0.633 - 0.773i$
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: \((2,\ 10,\ (\ :9),\ 0.633 - 0.773i)\)

Particular Values

\(L(\frac{19}{2})\) \(\approx\) \(3.23461 + 1.53131i\)
\(L(\frac12)\) \(\approx\) \(3.23461 + 1.53131i\)
\(L(10)\) 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 \( 1 + (-1.75e6 - 8.57e5i)T \)
good3 \( 1 + (-1.07e4 + 1.07e4i)T - 3.87e8iT^{2} \)
7 \( 1 + (4.02e6 + 4.02e6i)T + 1.62e15iT^{2} \)
11 \( 1 - 1.03e9T + 5.55e18T^{2} \)
13 \( 1 + (-6.67e9 + 6.67e9i)T - 1.12e20iT^{2} \)
17 \( 1 + (-7.11e10 - 7.11e10i)T + 1.40e22iT^{2} \)
19 \( 1 - 2.17e11iT - 1.04e23T^{2} \)
23 \( 1 + (-5.11e11 + 5.11e11i)T - 3.24e24iT^{2} \)
29 \( 1 - 1.96e13iT - 2.10e26T^{2} \)
31 \( 1 - 1.66e13T + 6.99e26T^{2} \)
37 \( 1 + (1.60e14 + 1.60e14i)T + 1.68e28iT^{2} \)
41 \( 1 + 3.31e13T + 1.07e29T^{2} \)
43 \( 1 + (-3.83e14 + 3.83e14i)T - 2.52e29iT^{2} \)
47 \( 1 + (1.26e15 + 1.26e15i)T + 1.25e30iT^{2} \)
53 \( 1 + (9.46e14 - 9.46e14i)T - 1.08e31iT^{2} \)
59 \( 1 + 4.59e15iT - 7.50e31T^{2} \)
61 \( 1 - 1.10e16T + 1.36e32T^{2} \)
67 \( 1 + (2.22e16 + 2.22e16i)T + 7.40e32iT^{2} \)
71 \( 1 + 8.76e16T + 2.10e33T^{2} \)
73 \( 1 + (4.04e16 - 4.04e16i)T - 3.46e33iT^{2} \)
79 \( 1 + 3.77e16iT - 1.43e34T^{2} \)
83 \( 1 + (-3.05e15 + 3.05e15i)T - 3.49e34iT^{2} \)
89 \( 1 - 5.34e17iT - 1.22e35T^{2} \)
97 \( 1 + (3.98e17 + 3.98e17i)T + 5.77e35iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.48914813276283167772358064011, −14.69811866737810378198701838844, −13.78390200390882951812867272385, −12.67965766694055894054157410289, −10.49869023946765258046135763972, −8.535757769808206309489024719681, −7.00379254475871130380939777917, −5.56829459725129267018102405903, −3.29584150903253570374580553203, −1.68559009823561262868222672472, 1.20567776125474188449868102643, 2.92911235802281792426875678798, 4.51378233194787316151908876505, 6.23796228631994224533323311570, 8.964630293920342511536207366958, 9.903883366678829722580129588641, 11.77814272425825476459012534841, 13.38557212164095305674212147485, 14.42438670623366151298193444818, 15.88370602795031813518112957166

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