↑

Properties

Label 2-975-195.44-c1-0-39
Degree $2$
Conductor $975$
Sign $0.660 + 0.750i$
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  + (−0.789 + 0.789i)2-s + (−0.548 − 1.64i)3-s + 0.754i·4-s + (1.72 + 0.863i)6-s + (−1.97 + 1.97i)7-s + (−2.17 − 2.17i)8-s + (−2.39 + 1.80i)9-s + (−3.56 + 3.56i)11-s + (1.23 − 0.413i)12-s + (1.26 − 3.37i)13-s − 3.12i·14-s + 1.92·16-s + 0.700i·17-s + (0.470 − 3.31i)18-s + (4.32 − 4.32i)19-s + ⋯
L(s)  = 1  + (−0.558 + 0.558i)2-s + (−0.316 − 0.948i)3-s + 0.377i·4-s + (0.706 + 0.352i)6-s + (−0.747 + 0.747i)7-s + (−0.768 − 0.768i)8-s + (−0.799 + 0.600i)9-s + (−1.07 + 1.07i)11-s + (0.357 − 0.119i)12-s + (0.350 − 0.936i)13-s − 0.834i·14-s + 0.480·16-s + 0.169i·17-s + (0.110 − 0.781i)18-s + (0.992 − 0.992i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.660 + 0.750i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.660 + 0.750i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (824, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.660 + 0.750i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.513894 - 0.232178i\)
\(L(\frac12)\) \(\approx\) \(0.513894 - 0.232178i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.548 + 1.64i)T \)
5 \( 1 \)
13 \( 1 + (-1.26 + 3.37i)T \)
good2 \( 1 + (0.789 - 0.789i)T - 2iT^{2} \)
7 \( 1 + (1.97 - 1.97i)T - 7iT^{2} \)
11 \( 1 + (3.56 - 3.56i)T - 11iT^{2} \)
17 \( 1 - 0.700iT - 17T^{2} \)
19 \( 1 + (-4.32 + 4.32i)T - 19iT^{2} \)
23 \( 1 + 4.02iT - 23T^{2} \)
29 \( 1 - 6.96iT - 29T^{2} \)
31 \( 1 + (-6.53 + 6.53i)T - 31iT^{2} \)
37 \( 1 + (-2.65 + 2.65i)T - 37iT^{2} \)
41 \( 1 + (5.37 + 5.37i)T + 41iT^{2} \)
43 \( 1 - 1.62T + 43T^{2} \)
47 \( 1 + (3.82 + 3.82i)T + 47iT^{2} \)
53 \( 1 - 0.258T + 53T^{2} \)
59 \( 1 + (-9.68 + 9.68i)T - 59iT^{2} \)
61 \( 1 + 3.98T + 61T^{2} \)
67 \( 1 + (4.24 + 4.24i)T + 67iT^{2} \)
71 \( 1 + (-2.54 - 2.54i)T + 71iT^{2} \)
73 \( 1 + (6.24 - 6.24i)T - 73iT^{2} \)
79 \( 1 + 0.921T + 79T^{2} \)
83 \( 1 + (-8.89 + 8.89i)T - 83iT^{2} \)
89 \( 1 + (-9.13 + 9.13i)T - 89iT^{2} \)
97 \( 1 + (-1.54 - 1.54i)T + 97iT^{2} \)
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

−9.729153037927126923395997201059, −8.818703537955540699669625927939, −8.091249983793194765454085706367, −7.38556883856194604517450844307, −6.71508548180099053354301190685, −5.85242800637255582276665632820, −4.95612460988569932463179475962, −3.16902541198570713836361695062, −2.42463643290839130633052040324, −0.40306470902704152899193847504, 1.00757493670575307751526320887, 2.85064235305645622473223476134, 3.64314265163335731502706251771, 4.88527121499130124242213104102, 5.80277840171003800925317305099, 6.46143130912165544216251758863, 7.899717764092744534589724381169, 8.747869618331678935337675175554, 9.670759134339432481877694751401, 10.06669948827727602698004155906

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