↑

Properties

Label 2-525-35.4-c1-0-11
Degree $2$
Conductor $525$
Sign $0.943 + 0.330i$
Analytic cond. $4.19214$
Root an. cond. $2.04747$
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  + (−0.866 + 0.5i)2-s + (−0.866 − 0.5i)3-s + (−0.500 + 0.866i)4-s + 0.999·6-s + (−2.59 + 0.5i)7-s − 3i·8-s + (0.499 + 0.866i)9-s + (0.866 − 0.499i)12-s − 3i·13-s + (2 − 1.73i)14-s + (0.500 + 0.866i)16-s + (1.73 + i)17-s + (−0.866 − 0.499i)18-s + (0.5 + 0.866i)19-s + (2.5 + 0.866i)21-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.499 − 0.288i)3-s + (−0.250 + 0.433i)4-s + 0.408·6-s + (−0.981 + 0.188i)7-s − 1.06i·8-s + (0.166 + 0.288i)9-s + (0.249 − 0.144i)12-s − 0.832i·13-s + (0.534 − 0.462i)14-s + (0.125 + 0.216i)16-s + (0.420 + 0.242i)17-s + (−0.204 − 0.117i)18-s + (0.114 + 0.198i)19-s + (0.545 + 0.188i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.330i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 + 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.943 + 0.330i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (424, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.943 + 0.330i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.607371 - 0.103289i\)
\(L(\frac12)\) \(\approx\) \(0.607371 - 0.103289i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 \)
7 \( 1 + (2.59 - 0.5i)T \)
good2 \( 1 + (0.866 - 0.5i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 3iT - 13T^{2} \)
17 \( 1 + (-1.73 - i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.73 + i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 8T + 29T^{2} \)
31 \( 1 + (-4 + 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.06 - 3.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 + (-8.66 + 5i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-12.1 - 7i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5 + 8.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.33 + 2.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + (-9.52 - 5.5i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.5 + 6.06i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 14iT - 83T^{2} \)
89 \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 9iT - 97T^{2} \)
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

−10.44003703966243138540939566779, −9.999899610923579555365357962510, −8.915272957681250639932330361441, −8.147753213175085750875338697775, −7.20091891996140394553794202949, −6.42751053576422458762043167726, −5.43073996210336819989443082228, −4.00581939954847360424214412817, −2.85955705465982351805212457988, −0.61682798557765087417713721348, 1.04873917174459805634017296509, 2.82094220221747914281718755376, 4.27367659685432180208384723539, 5.30335368193186599788513691409, 6.31231722360307485970168648912, 7.19867953340156791415853024226, 8.653757824385619702709879140622, 9.267731846835761147476772485418, 10.16048045320879872204687234652, 10.56081814475213856788456711549

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