↑

Properties

Label 2-975-65.29-c1-0-24
Degree $2$
Conductor $975$
Sign $0.900 - 0.435i$
Analytic cond. $7.78541$
Root an. cond. $2.79023$
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  + (−1.73 + i)2-s + (0.866 − 0.5i)3-s + (0.999 − 1.73i)4-s + (−0.999 + 1.73i)6-s + (4.33 + 2.5i)7-s + (0.499 − 0.866i)9-s + (−1 − 1.73i)11-s − 1.99i·12-s + (2.59 − 2.5i)13-s − 10·14-s + (1.99 + 3.46i)16-s + (1.73 + i)17-s + 1.99i·18-s + 5·21-s + (3.46 + 1.99i)22-s + (5.19 − 3i)23-s + ⋯
L(s)  = 1  + (−1.22 + 0.707i)2-s + (0.499 − 0.288i)3-s + (0.499 − 0.866i)4-s + (−0.408 + 0.707i)6-s + (1.63 + 0.944i)7-s + (0.166 − 0.288i)9-s + (−0.301 − 0.522i)11-s − 0.577i·12-s + (0.720 − 0.693i)13-s − 2.67·14-s + (0.499 + 0.866i)16-s + (0.420 + 0.242i)17-s + 0.471i·18-s + 1.09·21-s + (0.738 + 0.426i)22-s + (1.08 − 0.625i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 - 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.900 - 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.900 - 0.435i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (874, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.900 - 0.435i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20871 + 0.277169i\)
\(L(\frac12)\) \(\approx\) \(1.20871 + 0.277169i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 + (-2.59 + 2.5i)T \)
good2 \( 1 + (1.73 - i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (-4.33 - 2.5i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.73 - i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.19 + 3i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2 + 3.46i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 7T + 31T^{2} \)
37 \( 1 + (-1.73 + i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (3 + 5.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.866 + 0.5i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 8iT - 47T^{2} \)
53 \( 1 + 4iT - 53T^{2} \)
59 \( 1 + (-6 + 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.5 + 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.06 + 3.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (6 - 10.3i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 15iT - 73T^{2} \)
79 \( 1 + 3T + 79T^{2} \)
83 \( 1 - 8iT - 83T^{2} \)
89 \( 1 + (-7 - 12.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.33 + 2.5i)T + (48.5 + 84.0i)T^{2} \)
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

−9.725168038810136099225983965140, −8.857341686949748874432439732037, −8.303817809762040164020229679250, −7.972664936406282553442611060227, −7.03254375715097706848105471158, −5.90354986920880890578635481719, −5.17552345217781984027073613392, −3.65464600933144177021235355474, −2.21943833839312897957195844635, −1.03915961542936239534886886246, 1.23627388046903148292952137294, 1.96519662786247475715026195996, 3.40005673187505533327921984541, 4.53427654417164607186827999119, 5.35950304284896000498866983922, 7.26545674582985427115708487986, 7.56021108252915700427688254229, 8.599688419793462980309731055271, 9.004086425312652713517686711554, 10.05691524615479795710762251856

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