Properties

Label 2-725-5.4-c1-0-35
Degree $2$
Conductor $725$
Sign $-0.447 + 0.894i$
Analytic cond. $5.78915$
Root an. cond. $2.40606$
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  + 0.311i·2-s − 2.90i·3-s + 1.90·4-s + 0.903·6-s − 0.903i·7-s + 1.21i·8-s − 5.42·9-s − 1.52·11-s − 5.52i·12-s + 0.622i·13-s + 0.280·14-s + 3.42·16-s − 7.95i·17-s − 1.68i·18-s + 1.09·19-s + ⋯
L(s)  = 1  + 0.219i·2-s − 1.67i·3-s + 0.951·4-s + 0.368·6-s − 0.341i·7-s + 0.429i·8-s − 1.80·9-s − 0.459·11-s − 1.59i·12-s + 0.172i·13-s + 0.0750·14-s + 0.857·16-s − 1.92i·17-s − 0.398i·18-s + 0.251·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(725\)    =    \(5^{2} \cdot 29\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(5.78915\)
Root analytic conductor: \(2.40606\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{725} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 725,\ (\ :1/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.895550 - 1.44903i\)
\(L(\frac12)\) \(\approx\) \(0.895550 - 1.44903i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
29 \( 1 - T \)
good2 \( 1 - 0.311iT - 2T^{2} \)
3 \( 1 + 2.90iT - 3T^{2} \)
7 \( 1 + 0.903iT - 7T^{2} \)
11 \( 1 + 1.52T + 11T^{2} \)
13 \( 1 - 0.622iT - 13T^{2} \)
17 \( 1 + 7.95iT - 17T^{2} \)
19 \( 1 - 1.09T + 19T^{2} \)
23 \( 1 + 7.52iT - 23T^{2} \)
31 \( 1 + 6.90T + 31T^{2} \)
37 \( 1 - 3.95iT - 37T^{2} \)
41 \( 1 - 3.67T + 41T^{2} \)
43 \( 1 - 10.5iT - 43T^{2} \)
47 \( 1 - 6.90iT - 47T^{2} \)
53 \( 1 + 6.42iT - 53T^{2} \)
59 \( 1 - 1.67T + 59T^{2} \)
61 \( 1 + 1.86T + 61T^{2} \)
67 \( 1 - 11.5iT - 67T^{2} \)
71 \( 1 - 13.6T + 71T^{2} \)
73 \( 1 + 10.1iT - 73T^{2} \)
79 \( 1 + 9.13T + 79T^{2} \)
83 \( 1 + 10.7iT - 83T^{2} \)
89 \( 1 - 7.80T + 89T^{2} \)
97 \( 1 + 4.08iT - 97T^{2} \)
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

−10.29511142618957282033278725098, −9.027926668470267888693543204992, −7.933515043240444306715642632790, −7.42005009880733416807299668669, −6.77337134463805486377452422616, −6.05922060633589846569362859027, −4.93346181577913420301805047446, −2.96748894532238599423757763898, −2.23108985243050036002025773489, −0.860816226238208869574360757040, 2.04384151960401830788454413007, 3.38362902929736051252416140603, 3.95157569051803362898232991906, 5.43399060340317953303435568188, 5.82947200560650128798166384303, 7.24788657996021475512535112939, 8.282108493778085856537304040906, 9.197027215818209524997250729903, 10.03040429909510617737200092921, 10.69087726340978462433595202860

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