↑

Properties

Label 2-525-21.5-c1-0-41
Degree $2$
Conductor $525$
Sign $-0.444 + 0.895i$
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  + (1.5 − 0.866i)3-s + (−1 − 1.73i)4-s + (−0.5 − 2.59i)7-s + (1.5 − 2.59i)9-s + (−3 − 1.73i)12-s + 1.73i·13-s + (−1.99 + 3.46i)16-s + (−4.5 − 2.59i)19-s + (−3 − 3.46i)21-s − 5.19i·27-s + (−4 + 3.46i)28-s + (7.5 − 4.33i)31-s − 6·36-s + (0.5 − 0.866i)37-s + (1.49 + 2.59i)39-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s + (−0.5 − 0.866i)4-s + (−0.188 − 0.981i)7-s + (0.5 − 0.866i)9-s + (−0.866 − 0.499i)12-s + 0.480i·13-s + (−0.499 + 0.866i)16-s + (−1.03 − 0.596i)19-s + (−0.654 − 0.755i)21-s − 0.999i·27-s + (−0.755 + 0.654i)28-s + (1.34 − 0.777i)31-s − 36-s + (0.0821 − 0.142i)37-s + (0.240 + 0.416i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.444 + 0.895i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.444 + 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.807315 - 1.30127i\)
\(L(\frac12)\) \(\approx\) \(0.807315 - 1.30127i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.5 + 0.866i)T \)
5 \( 1 \)
7 \( 1 + (0.5 + 2.59i)T \)
good2 \( 1 + (1 + 1.73i)T^{2} \)
11 \( 1 + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.73iT - 13T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.5 + 2.59i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (-7.5 + 4.33i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 5T + 43T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6 - 3.46i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.5 - 9.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-13.5 + 7.79i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.5 + 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 13.8iT - 97T^{2} \)
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

−10.37650257749575262375681195705, −9.641709154406238161222607458249, −8.866237202378373240600551070480, −7.952681727012636919654588356996, −6.87867987815446069435897347762, −6.20240937006674184708994844485, −4.64585070953819687191436353598, −3.86116311364689938946201580604, −2.28147068853533253333925174862, −0.848079794222887665276495082755, 2.36153412573112177706269710039, 3.28687116431095061510459503434, 4.30830464324035628712094546409, 5.32965068511905031149613030409, 6.70720853015046547331354729237, 8.063727837778368209563984116739, 8.369414729250721848152321888788, 9.264492018743605027303909701158, 9.999621213421722491316626640805, 11.07819973987764280653510974994

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