↑

Properties

Label 2-725-145.109-c1-0-16
Degree $2$
Conductor $725$
Sign $0.936 - 0.349i$
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  + (0.0209 + 0.0919i)2-s + (0.209 + 0.100i)3-s + (1.79 − 0.863i)4-s + (−0.00487 + 0.0213i)6-s + (−1.49 + 3.10i)7-s + (0.234 + 0.294i)8-s + (−1.83 − 2.30i)9-s + (−0.00755 − 0.00602i)11-s + 0.462·12-s + (3.63 + 2.90i)13-s + (−0.317 − 0.0724i)14-s + (2.46 − 3.08i)16-s + 5.88·17-s + (0.173 − 0.217i)18-s + (2.32 + 4.83i)19-s + ⋯
L(s)  = 1  + (0.0148 + 0.0650i)2-s + (0.120 + 0.0581i)3-s + (0.896 − 0.431i)4-s + (−0.00198 + 0.00871i)6-s + (−0.565 + 1.17i)7-s + (0.0830 + 0.104i)8-s + (−0.612 − 0.767i)9-s + (−0.00227 − 0.00181i)11-s + 0.133·12-s + (1.00 + 0.804i)13-s + (−0.0847 − 0.0193i)14-s + (0.615 − 0.771i)16-s + 1.42·17-s + (0.0408 − 0.0512i)18-s + (0.534 + 1.10i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.349i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(725\)    =    \(5^{2} \cdot 29\)
Sign: $0.936 - 0.349i$
Analytic conductor: \(5.78915\)
Root analytic conductor: \(2.40606\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{725} (399, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 725,\ (\ :1/2),\ 0.936 - 0.349i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.89796 + 0.342645i\)
\(L(\frac12)\) \(\approx\) \(1.89796 + 0.342645i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.15 - 3.42i)T \)
good2 \( 1 + (-0.0209 - 0.0919i)T + (-1.80 + 0.867i)T^{2} \)
3 \( 1 + (-0.209 - 0.100i)T + (1.87 + 2.34i)T^{2} \)
7 \( 1 + (1.49 - 3.10i)T + (-4.36 - 5.47i)T^{2} \)
11 \( 1 + (0.00755 + 0.00602i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (-3.63 - 2.90i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 - 5.88T + 17T^{2} \)
19 \( 1 + (-2.32 - 4.83i)T + (-11.8 + 14.8i)T^{2} \)
23 \( 1 + (-3.11 - 0.711i)T + (20.7 + 9.97i)T^{2} \)
31 \( 1 + (-9.42 + 2.15i)T + (27.9 - 13.4i)T^{2} \)
37 \( 1 + (5.25 + 6.59i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 + 7.41iT - 41T^{2} \)
43 \( 1 + (-0.472 + 2.07i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (2.03 - 2.55i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (9.49 - 2.16i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + 0.911T + 59T^{2} \)
61 \( 1 + (2.97 - 6.17i)T + (-38.0 - 47.6i)T^{2} \)
67 \( 1 + (6.75 - 5.38i)T + (14.9 - 65.3i)T^{2} \)
71 \( 1 + (-9.02 + 11.3i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (2.73 - 11.9i)T + (-65.7 - 31.6i)T^{2} \)
79 \( 1 + (-1.68 + 1.34i)T + (17.5 - 77.0i)T^{2} \)
83 \( 1 + (3.03 + 6.29i)T + (-51.7 + 64.8i)T^{2} \)
89 \( 1 + (0.908 - 0.207i)T + (80.1 - 38.6i)T^{2} \)
97 \( 1 + (9.49 - 4.57i)T + (60.4 - 75.8i)T^{2} \)
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

−10.42906712167147766344851784641, −9.505226265364634445893829926373, −8.898250556458555925761444117301, −7.85648460117458077025896454837, −6.74349593616857949168947819641, −5.90616195263057789854832344616, −5.52856531125657252528623832419, −3.62541039377554387879557027654, −2.84241729115902163560407365605, −1.44646997926465291734117848173, 1.16161780848240245174230563865, 2.94299005321046625296145079123, 3.41150264434949549716699942408, 4.90544429144498377299840696577, 6.08920454088446999257168708367, 6.90682704555453417548909089690, 7.82209709615954298659063912677, 8.282305501131063537343638269565, 9.716576526017867998231257897534, 10.49849061765529039716577479444

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