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Properties

Label 2-975-65.29-c1-0-20
Degree $2$
Conductor $975$
Sign $0.999 - 0.0338i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.95 + 1.13i)2-s + (0.866 − 0.5i)3-s + (1.55 − 2.69i)4-s + (−1.13 + 1.95i)6-s + (1.09 + 0.630i)7-s + 2.52i·8-s + (0.499 − 0.866i)9-s + (−2.26 − 3.91i)11-s − 3.11i·12-s + (−1.04 + 3.45i)13-s − 2.85·14-s + (0.261 + 0.453i)16-s + (3.89 + 2.24i)17-s + 2.26i·18-s + (2.55 − 4.43i)19-s + ⋯
L(s)  = 1  + (−1.38 + 0.799i)2-s + (0.499 − 0.288i)3-s + (0.778 − 1.34i)4-s + (−0.461 + 0.799i)6-s + (0.413 + 0.238i)7-s + 0.892i·8-s + (0.166 − 0.288i)9-s + (−0.681 − 1.18i)11-s − 0.899i·12-s + (−0.290 + 0.957i)13-s − 0.762·14-s + (0.0654 + 0.113i)16-s + (0.943 + 0.544i)17-s + 0.533i·18-s + (0.586 − 1.01i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0338i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0338i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.999 - 0.0338i$
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: \((2,\ 975,\ (\ :1/2),\ 0.999 - 0.0338i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.926207 + 0.0157036i\)
\(L(\frac12)\) \(\approx\) \(0.926207 + 0.0157036i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.04 - 3.45i)T \)
good2 \( 1 + (1.95 - 1.13i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (-1.09 - 0.630i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.26 + 3.91i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3.89 - 2.24i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.55 + 4.43i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.93 - 1.11i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.688 + 1.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 8.87T + 31T^{2} \)
37 \( 1 + (0.200 - 0.115i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.573 + 0.992i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.52 - 3.18i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 10.7iT - 47T^{2} \)
53 \( 1 + 4.52iT - 53T^{2} \)
59 \( 1 + (0.426 - 0.739i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.31 - 4.01i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-11.3 + 6.56i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.80 + 8.32i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 13.7iT - 73T^{2} \)
79 \( 1 - 8.87T + 79T^{2} \)
83 \( 1 - 8.23iT - 83T^{2} \)
89 \( 1 + (-3.31 - 5.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (9.24 + 5.33i)T + (48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.690666775820429188597969573340, −9.013603260256943694126083743793, −8.201146645021035344571061637191, −7.85760648287486822162236488399, −6.86380204583698965041225253297, −6.09346699706484459997415540367, −5.06659053187449910599338107785, −3.50665041554147398348997959452, −2.15609213310718831549318228313, −0.78701785559276411379273903530, 1.11585270968088732754125548874, 2.36641187441258732899647569051, 3.18584403962802770143066348153, 4.56132162555688965208017254911, 5.57772782193859122261734237441, 7.27279823542227429693665638824, 7.85181800874882457352630931915, 8.282519519657676162610997885203, 9.517823552427104510747201851120, 9.953256489256796079045395616053

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