Properties

Label 2-525-105.23-c1-0-14
Degree $2$
Conductor $525$
Sign $0.892 - 0.450i$
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  + (0.448 − 1.67i)3-s + (−1.73 + i)4-s + (0.189 + 2.63i)7-s + (−2.59 − 1.50i)9-s + (0.896 + 3.34i)12-s + (4.89 + 4.89i)13-s + (1.99 − 3.46i)16-s + (6.92 + 4i)19-s + (4.50 + 0.866i)21-s + (−3.67 + 3.67i)27-s + (−2.96 − 4.38i)28-s + (−3.5 − 6.06i)31-s + 6·36-s + (1.34 + 5.01i)37-s + (10.3 − 6i)39-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)3-s + (−0.866 + 0.5i)4-s + (0.0716 + 0.997i)7-s + (−0.866 − 0.5i)9-s + (0.258 + 0.965i)12-s + (1.35 + 1.35i)13-s + (0.499 − 0.866i)16-s + (1.58 + 0.917i)19-s + (0.981 + 0.188i)21-s + (−0.707 + 0.707i)27-s + (−0.560 − 0.827i)28-s + (−0.628 − 1.08i)31-s + 36-s + (0.221 + 0.825i)37-s + (1.66 − 0.960i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 - 0.450i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.892 - 0.450i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.892 - 0.450i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.892 - 0.450i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23831 + 0.294879i\)
\(L(\frac12)\) \(\approx\) \(1.23831 + 0.294879i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.448 + 1.67i)T \)
5 \( 1 \)
7 \( 1 + (-0.189 - 2.63i)T \)
good2 \( 1 + (1.73 - i)T^{2} \)
11 \( 1 + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.89 - 4.89i)T + 13iT^{2} \)
17 \( 1 + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-6.92 - 4i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (3.5 + 6.06i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.34 - 5.01i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (-8.57 - 8.57i)T + 43iT^{2} \)
47 \( 1 + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.34 - 0.896i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (0.448 - 1.67i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (14.7 + 8.5i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.67 + 3.67i)T - 97iT^{2} \)
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

−11.38420091019427001481609761423, −9.592972153111262260018271617719, −9.056286482656038004168503835863, −8.268195537121943742124075619364, −7.53812592438099509352609599492, −6.28836015723211368458602126801, −5.50487132403223815033590005044, −4.05567267908212512396076764298, −2.97219976236701242639700440210, −1.46472020393295575028365369147, 0.866404681991786976776496362435, 3.25177775853202690774510881919, 4.00691147074192089685643714493, 5.11825562724216682660506086223, 5.77391138178279621155975838917, 7.34668101451046793403080864168, 8.368308989940967166818592667592, 9.112439107446540772243682580315, 9.957131323603720704325811845198, 10.68597247580603178185822719699

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