Properties

Label 2-325-13.11-c2-0-19
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

\(L(\frac{3}{2})\) \(\approx\) \(1.87279 + 0.127640i\)
\(L(\frac12)\) \(\approx\) \(1.87279 + 0.127640i\)
\(L(2)\) 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

−11.39435533734420701360857407917, −10.68886997442873698625982934902, −9.358061904524415754498004954338, −8.545579074361790945067027551987, −7.13424587315311707837856756015, −6.90993998919043649134770315814, −5.81040723843897470272980904606, −4.16681938832502895371449439714, −3.00746545740143114574549969006, −1.15063476157722674513983717532, 1.37850040041405819033443726482, 2.68908971851571307485892957502, 4.10536260102852787416827696662, 5.56706413028283887199633680735, 6.35755702129811278335497578892, 7.48588770071422031749839312513, 8.774071232806221087868222026606, 9.653785936365996809251945762406, 10.60112970484190878286139554892, 11.27276650249275144494371627479

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