↑

Properties

Label 2-10e3-25.21-c1-0-16
Degree $2$
Conductor $1000$
Sign $0.506 + 0.862i$
Analytic cond. $7.98504$
Root an. cond. $2.82578$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.0481 + 0.0350i)3-s + 1.71·7-s + (−0.925 − 2.84i)9-s + (0.573 − 1.76i)11-s + (0.173 + 0.533i)13-s + (3.15 − 2.29i)17-s + (−4.69 + 3.40i)19-s + (0.0827 + 0.0601i)21-s + (2.25 − 6.95i)23-s + (0.110 − 0.339i)27-s + (−2.13 − 1.55i)29-s + (1.56 − 1.13i)31-s + (0.0894 − 0.0649i)33-s + (−0.235 − 0.725i)37-s + (−0.0103 + 0.0317i)39-s + ⋯
L(s)  = 1  + (0.0278 + 0.0202i)3-s + 0.648·7-s + (−0.308 − 0.949i)9-s + (0.172 − 0.532i)11-s + (0.0481 + 0.148i)13-s + (0.766 − 0.556i)17-s + (−1.07 + 0.782i)19-s + (0.0180 + 0.0131i)21-s + (0.471 − 1.45i)23-s + (0.0212 − 0.0653i)27-s + (−0.396 − 0.288i)29-s + (0.280 − 0.203i)31-s + (0.0155 − 0.0113i)33-s + (−0.0387 − 0.119i)37-s + (−0.00165 + 0.00509i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.506 + 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.506 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1000\)    =    \(2^{3} \cdot 5^{3}\)
Sign: $0.506 + 0.862i$
Analytic conductor: \(7.98504\)
Root analytic conductor: \(2.82578\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1000} (601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1000,\ (\ :1/2),\ 0.506 + 0.862i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38360 - 0.791991i\)
\(L(\frac12)\) \(\approx\) \(1.38360 - 0.791991i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.0481 - 0.0350i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 - 1.71T + 7T^{2} \)
11 \( 1 + (-0.573 + 1.76i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-0.173 - 0.533i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-3.15 + 2.29i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (4.69 - 3.40i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-2.25 + 6.95i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (2.13 + 1.55i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.56 + 1.13i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.235 + 0.725i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (3.00 + 9.25i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 9.68T + 43T^{2} \)
47 \( 1 + (-8.42 - 6.11i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.900 - 0.654i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.56 - 4.81i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-4.18 + 12.8i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-1.01 + 0.733i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (-10.8 - 7.91i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.26 + 3.88i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (5.25 + 3.81i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (0.738 - 0.536i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-1.61 + 4.97i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (11.7 + 8.54i)T + (29.9 + 92.2i)T^{2} \)
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

−9.803084412027427090026098916920, −8.880779762440146816452782451222, −8.344154520120481311959142424631, −7.33715345323707598198798529622, −6.34059723823085286170011262753, −5.64027153952577316609125415918, −4.45740397315326466710168164888, −3.57134750503801980285592270431, −2.34194976528930015022774024324, −0.78295242110199671940759766756, 1.49973653287605636770267328074, 2.61082852040625530998912352722, 3.95075406970584251942110705667, 4.95669401774688004466869410500, 5.64653801047681124811995076094, 6.85248505864229534340405271143, 7.70993853683715851345278217966, 8.344977206412235618825629193550, 9.245237350154945305950707296262, 10.19878204641928939083769821235

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