↑

Properties

Label 2-325-13.11-c2-0-18
Degree $2$
Conductor $325$
Sign $0.990 - 0.135i$
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  + (0.222 − 0.830i)2-s + (0.435 + 0.754i)3-s + (2.82 + 1.63i)4-s + (0.724 − 0.194i)6-s + (1.18 + 4.40i)7-s + (4.41 − 4.41i)8-s + (4.12 − 7.13i)9-s + (5.64 + 1.51i)11-s + 2.84i·12-s + (−12.9 + 1.26i)13-s + 3.92·14-s + (3.83 + 6.64i)16-s + (14.1 + 8.16i)17-s + (−5.01 − 5.01i)18-s + (17.2 − 4.62i)19-s + ⋯
L(s)  = 1  + (0.111 − 0.415i)2-s + (0.145 + 0.251i)3-s + (0.705 + 0.407i)4-s + (0.120 − 0.0323i)6-s + (0.168 + 0.629i)7-s + (0.551 − 0.551i)8-s + (0.457 − 0.792i)9-s + (0.513 + 0.137i)11-s + 0.236i·12-s + (−0.995 + 0.0973i)13-s + 0.280·14-s + (0.239 + 0.415i)16-s + (0.832 + 0.480i)17-s + (−0.278 − 0.278i)18-s + (0.908 − 0.243i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.135i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.990 - 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.990 - 0.135i$
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: \((2,\ 325,\ (\ :1),\ 0.990 - 0.135i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.37371 + 0.161780i\)
\(L(\frac12)\) \(\approx\) \(2.37371 + 0.161780i\)
\(L(2)\) 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 \( 1 + (12.9 - 1.26i)T \)
good2 \( 1 + (-0.222 + 0.830i)T + (-3.46 - 2i)T^{2} \)
3 \( 1 + (-0.435 - 0.754i)T + (-4.5 + 7.79i)T^{2} \)
7 \( 1 + (-1.18 - 4.40i)T + (-42.4 + 24.5i)T^{2} \)
11 \( 1 + (-5.64 - 1.51i)T + (104. + 60.5i)T^{2} \)
17 \( 1 + (-14.1 - 8.16i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-17.2 + 4.62i)T + (312. - 180.5i)T^{2} \)
23 \( 1 + (17.4 - 10.0i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-23.1 - 40.0i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (6.38 + 6.38i)T + 961iT^{2} \)
37 \( 1 + (2.47 + 0.662i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (-8.73 + 32.6i)T + (-1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (63.1 + 36.4i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (4.72 - 4.72i)T - 2.20e3iT^{2} \)
53 \( 1 - 11.7T + 2.80e3T^{2} \)
59 \( 1 + (-7.51 - 28.0i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (-28.4 + 49.2i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-5.20 + 19.4i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (117. - 31.3i)T + (4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (-24.8 + 24.8i)T - 5.32e3iT^{2} \)
79 \( 1 + 84.1T + 6.24e3T^{2} \)
83 \( 1 + (19.8 + 19.8i)T + 6.88e3iT^{2} \)
89 \( 1 + (116. + 31.1i)T + (6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (-80.6 + 21.6i)T + (8.14e3 - 4.70e3i)T^{2} \)
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

−11.82606121190708133053837597591, −10.41015518270900171016328257094, −9.729286653547577378623906829112, −8.693485130981921426810952932749, −7.47602058884585422239107096978, −6.70376275246581951415242945693, −5.38652722225309507799030101165, −3.97042940005936509741072465040, −2.99297678400894922242330741795, −1.57406035431373792483677368428, 1.30629053770174816686082938348, 2.69595404086727816514193246932, 4.45700790529341556977950042105, 5.47690574510711287423492549995, 6.67552436652642172533971527275, 7.50866770861325556320144375164, 8.079338126152995685315519622469, 9.852987143622174170130598367203, 10.25039218100637940946911887507, 11.47502904652081114807132435771

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