↑

Properties

Label 2-975-39.2-c1-0-63
Degree $2$
Conductor $975$
Sign $-0.984 - 0.173i$
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  + (0.619 − 2.31i)2-s + (−1.64 + 0.529i)3-s + (−3.23 − 1.86i)4-s + (0.202 + 4.14i)6-s + (1.36 − 0.366i)7-s + (−2.93 + 2.93i)8-s + (2.43 − 1.74i)9-s + (1.69 + 0.453i)11-s + (6.31 + 1.36i)12-s + (1.59 − 3.23i)13-s − 3.38i·14-s + (1.23 + 2.13i)16-s + (1.07 − 1.85i)17-s + (−2.52 − 6.72i)18-s + (−0.267 − i)19-s + ⋯
L(s)  = 1  + (0.438 − 1.63i)2-s + (−0.952 + 0.305i)3-s + (−1.61 − 0.933i)4-s + (0.0826 + 1.69i)6-s + (0.516 − 0.138i)7-s + (−1.03 + 1.03i)8-s + (0.813 − 0.582i)9-s + (0.510 + 0.136i)11-s + (1.82 + 0.394i)12-s + (0.443 − 0.896i)13-s − 0.904i·14-s + (0.308 + 0.533i)16-s + (0.260 − 0.450i)17-s + (−0.595 − 1.58i)18-s + (−0.0614 − 0.229i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.984 - 0.173i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (626, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.984 - 0.173i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.111152 + 1.27205i\)
\(L(\frac12)\) \(\approx\) \(0.111152 + 1.27205i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.64 - 0.529i)T \)
5 \( 1 \)
13 \( 1 + (-1.59 + 3.23i)T \)
good2 \( 1 + (-0.619 + 2.31i)T + (-1.73 - i)T^{2} \)
7 \( 1 + (-1.36 + 0.366i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-1.69 - 0.453i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.07 + 1.85i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.267 + i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.79 - 2.76i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.46 + 4.46i)T - 31iT^{2} \)
37 \( 1 + (-1.76 + 6.59i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.166 + 0.619i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (7.09 + 4.09i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.77 - 6.77i)T - 47iT^{2} \)
53 \( 1 + 4.62iT - 53T^{2} \)
59 \( 1 + (1.23 + 4.62i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.46 - 2.26i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (4.62 - 1.23i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-6.09 - 6.09i)T + 73iT^{2} \)
79 \( 1 - 2T + 79T^{2} \)
83 \( 1 + (1.23 + 1.23i)T + 83iT^{2} \)
89 \( 1 + (-9.70 - 2.60i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (3.36 + 12.5i)T + (-84.0 + 48.5i)T^{2} \)
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

−9.868511893318784563470067873503, −9.305632264642221636840344909635, −8.050297914394109398906685133447, −6.83927610173968540387731621706, −5.66170608872317079182250533371, −4.94187467929808046938137939556, −4.08929143065657439420014134158, −3.23007673195069516170447722880, −1.76962205649823639329677961332, −0.63047346395045924380227730113, 1.58179304624548669629488222262, 3.82687835903833832072624660752, 4.70790427205578621789388745367, 5.41401268170628632923993842615, 6.43568871518312286252944190486, 6.63064142019416988123517779286, 7.78651081353777475172318062534, 8.347729107367058337048067613983, 9.357736150964775103387402311568, 10.44613056703133197584818243173

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