↑

Properties

Label 2-325-13.8-c2-0-21
Degree $2$
Conductor $325$
Sign $-0.272 - 0.962i$
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  + (2.28 + 2.28i)2-s + 3.77·3-s + 6.47i·4-s + (8.64 + 8.64i)6-s + (−4.50 + 4.50i)7-s + (−5.66 + 5.66i)8-s + 5.26·9-s + (5.65 − 5.65i)11-s + 24.4i·12-s + (7.01 + 10.9i)13-s − 20.6·14-s − 0.0412·16-s + 15.0i·17-s + (12.0 + 12.0i)18-s + (−16.2 − 16.2i)19-s + ⋯
L(s)  = 1  + (1.14 + 1.14i)2-s + 1.25·3-s + 1.61i·4-s + (1.44 + 1.44i)6-s + (−0.643 + 0.643i)7-s + (−0.708 + 0.708i)8-s + 0.585·9-s + (0.514 − 0.514i)11-s + 2.03i·12-s + (0.539 + 0.842i)13-s − 1.47·14-s − 0.00257·16-s + 0.884i·17-s + (0.669 + 0.669i)18-s + (−0.853 − 0.853i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.272 - 0.962i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.272 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.272 - 0.962i$
Analytic conductor: \(8.85560\)
Root analytic conductor: \(2.97583\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1),\ -0.272 - 0.962i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.52632 + 3.33974i\)
\(L(\frac12)\) \(\approx\) \(2.52632 + 3.33974i\)
\(L(2)\) 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 \( 1 + (-7.01 - 10.9i)T \)
good2 \( 1 + (-2.28 - 2.28i)T + 4iT^{2} \)
3 \( 1 - 3.77T + 9T^{2} \)
7 \( 1 + (4.50 - 4.50i)T - 49iT^{2} \)
11 \( 1 + (-5.65 + 5.65i)T - 121iT^{2} \)
17 \( 1 - 15.0iT - 289T^{2} \)
19 \( 1 + (16.2 + 16.2i)T + 361iT^{2} \)
23 \( 1 + 29.6iT - 529T^{2} \)
29 \( 1 - 47.7T + 841T^{2} \)
31 \( 1 + (37.6 + 37.6i)T + 961iT^{2} \)
37 \( 1 + (-30.0 + 30.0i)T - 1.36e3iT^{2} \)
41 \( 1 + (-7.53 - 7.53i)T + 1.68e3iT^{2} \)
43 \( 1 + 14.7iT - 1.84e3T^{2} \)
47 \( 1 + (-16.6 + 16.6i)T - 2.20e3iT^{2} \)
53 \( 1 + 35.3T + 2.80e3T^{2} \)
59 \( 1 + (-5.93 + 5.93i)T - 3.48e3iT^{2} \)
61 \( 1 + 119.T + 3.72e3T^{2} \)
67 \( 1 + (11.3 + 11.3i)T + 4.48e3iT^{2} \)
71 \( 1 + (37.2 + 37.2i)T + 5.04e3iT^{2} \)
73 \( 1 + (-74.6 + 74.6i)T - 5.32e3iT^{2} \)
79 \( 1 + 40.7T + 6.24e3T^{2} \)
83 \( 1 + (-28.9 - 28.9i)T + 6.88e3iT^{2} \)
89 \( 1 + (60.8 - 60.8i)T - 7.92e3iT^{2} \)
97 \( 1 + (-79.7 - 79.7i)T + 9.40e3iT^{2} \)
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

−12.17408608576579312004963892648, −10.80584680193661887399268081023, −9.225350253308384301249563388378, −8.706199998427932915225976534690, −7.80311858648886750698618315366, −6.49438439118263406686722881344, −6.08969719278067876072567909184, −4.46312751683136813312021858678, −3.62774881998559956930734450136, −2.46583097693498556916058786010, 1.48972972113455250889065362048, 2.92655111916440478815742123466, 3.52116081882655194078854873452, 4.52568804982369922791255614740, 5.93003370172477135082640248024, 7.29248791863260322294631426583, 8.413929959745858954755252187072, 9.580295468391940987456886220809, 10.24478437910552612467427783636, 11.21927806745415689216939769118

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