↑

Properties

Label 2-525-105.2-c1-0-39
Degree $2$
Conductor $525$
Sign $-0.819 + 0.573i$
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  + (−0.391 − 1.46i)2-s + (1.50 − 0.852i)3-s + (−0.246 + 0.142i)4-s + (−1.83 − 1.86i)6-s + (1.17 − 2.36i)7-s + (−1.83 − 1.83i)8-s + (1.54 − 2.57i)9-s + (0.791 − 0.457i)11-s + (−0.250 + 0.425i)12-s + (−3.07 + 3.07i)13-s + (−3.92 − 0.791i)14-s + (−2.24 + 3.88i)16-s + (1.16 + 0.311i)17-s + (−4.35 − 1.24i)18-s + (5.95 + 3.43i)19-s + ⋯
L(s)  = 1  + (−0.276 − 1.03i)2-s + (0.870 − 0.492i)3-s + (−0.123 + 0.0712i)4-s + (−0.749 − 0.762i)6-s + (0.444 − 0.895i)7-s + (−0.648 − 0.648i)8-s + (0.514 − 0.857i)9-s + (0.238 − 0.137i)11-s + (−0.0723 + 0.122i)12-s + (−0.854 + 0.854i)13-s + (−1.04 − 0.211i)14-s + (−0.561 + 0.971i)16-s + (0.281 + 0.0755i)17-s + (−1.02 − 0.294i)18-s + (1.36 + 0.788i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 + 0.573i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.819 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.819 + 0.573i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.819 + 0.573i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.536839 - 1.70262i\)
\(L(\frac12)\) \(\approx\) \(0.536839 - 1.70262i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.50 + 0.852i)T \)
5 \( 1 \)
7 \( 1 + (-1.17 + 2.36i)T \)
good2 \( 1 + (0.391 + 1.46i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (-0.791 + 0.457i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.07 - 3.07i)T - 13iT^{2} \)
17 \( 1 + (-1.16 - 0.311i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-5.95 - 3.43i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.88 - 0.505i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 2.72T + 29T^{2} \)
31 \( 1 + (2.31 + 4.01i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.774 + 0.207i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 0.922iT - 41T^{2} \)
43 \( 1 + (-4.80 + 4.80i)T - 43iT^{2} \)
47 \( 1 + (-2.71 - 10.1i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.85 - 10.6i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-4.94 - 8.55i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.533 + 0.924i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.83 + 6.83i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 0.557iT - 71T^{2} \)
73 \( 1 + (2.10 + 0.564i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-2.62 - 1.51i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.38 + 2.38i)T + 83iT^{2} \)
89 \( 1 + (-5.64 + 9.78i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.58 - 1.58i)T + 97iT^{2} \)
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

−10.42432214022094520201896111711, −9.645411894059818141197480127129, −9.067598854708010305895265922650, −7.70189361141775060831632730341, −7.24014502916172262619688004098, −5.99945947263656311111510857383, −4.25639208806058320065668293796, −3.38883008704930168386997971718, −2.14853855383318016507716109451, −1.12870363323242902377442581924, 2.29729788745803747606537981286, 3.26960159101599089576349683395, 4.97746383292008867147813641219, 5.53564207580454132804991419985, 6.97892051021222714444898025824, 7.72450261752988322351061925004, 8.407292593646007097947905360227, 9.239138010764439365737657678043, 9.913710427129041955027618517762, 11.20596860477973932074837370390

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