↑

Properties

Label 2-325-13.11-c2-0-39
Degree $2$
Conductor $325$
Sign $0.494 - 0.869i$
Analytic cond. $8.85560$
Root an. cond. $2.97583$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.00 − 3.73i)2-s + (−1.63 − 2.83i)3-s + (−9.46 − 5.46i)4-s + (−12.2 + 3.27i)6-s + (−2.30 − 8.61i)7-s + (−18.9 + 18.9i)8-s + (−0.844 + 1.46i)9-s + (3.75 + 1.00i)11-s + 35.7i·12-s + (6.85 − 11.0i)13-s − 34.4·14-s + (29.9 + 51.7i)16-s + (5.00 + 2.89i)17-s + (4.61 + 4.61i)18-s + (16.7 − 4.48i)19-s + ⋯
L(s)  = 1  + (0.500 − 1.86i)2-s + (−0.544 − 0.943i)3-s + (−2.36 − 1.36i)4-s + (−2.03 + 0.545i)6-s + (−0.329 − 1.23i)7-s + (−2.36 + 2.36i)8-s + (−0.0938 + 0.162i)9-s + (0.341 + 0.0913i)11-s + 2.97i·12-s + (0.527 − 0.849i)13-s − 2.46·14-s + (1.86 + 3.23i)16-s + (0.294 + 0.170i)17-s + (0.256 + 0.256i)18-s + (0.881 − 0.236i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.494 - 0.869i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.494 - 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.494 - 0.869i$
Analytic conductor: \(8.85560\)
Root analytic conductor: \(2.97583\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1),\ 0.494 - 0.869i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.05289 + 0.612738i\)
\(L(\frac12)\) \(\approx\) \(1.05289 + 0.612738i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 + (-6.85 + 11.0i)T \)
good2 \( 1 + (-1.00 + 3.73i)T + (-3.46 - 2i)T^{2} \)
3 \( 1 + (1.63 + 2.83i)T + (-4.5 + 7.79i)T^{2} \)
7 \( 1 + (2.30 + 8.61i)T + (-42.4 + 24.5i)T^{2} \)
11 \( 1 + (-3.75 - 1.00i)T + (104. + 60.5i)T^{2} \)
17 \( 1 + (-5.00 - 2.89i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-16.7 + 4.48i)T + (312. - 180.5i)T^{2} \)
23 \( 1 + (-8.48 + 4.89i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (7.28 + 12.6i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-26.8 - 26.8i)T + 961iT^{2} \)
37 \( 1 + (40.8 + 10.9i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (-11.2 + 42.0i)T + (-1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (-51.5 - 29.7i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (50.4 - 50.4i)T - 2.20e3iT^{2} \)
53 \( 1 + 28.3T + 2.80e3T^{2} \)
59 \( 1 + (2.48 + 9.25i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (-10.4 + 18.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (2.88 - 10.7i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (7.29 - 1.95i)T + (4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (-36.6 + 36.6i)T - 5.32e3iT^{2} \)
79 \( 1 + 45.8T + 6.24e3T^{2} \)
83 \( 1 + (-42.4 - 42.4i)T + 6.88e3iT^{2} \)
89 \( 1 + (86.1 + 23.0i)T + (6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (-48.7 + 13.0i)T + (8.14e3 - 4.70e3i)T^{2} \)
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

−10.82561326454778555589581157698, −10.13728768824812075575383158247, −9.187038808426098608973691968767, −7.76674217784408292570601092612, −6.48854902409581272510289115486, −5.33913602945890729555400986464, −4.05427135023479931158902279608, −3.10908253696449507129577855403, −1.38386279758750875707848285031, −0.60249262296732552721263320512, 3.46368385088013956676705645321, 4.56784644514338200672480333368, 5.45291358947692300424119740581, 6.07974041883225270963937477796, 7.12175465909076044142446013140, 8.334485790104927962151613589451, 9.191677012137646581700434110077, 9.799159600675121754310611606168, 11.45987253836553037457325218820, 12.27459365503071133461130995160

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