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Properties

Label 2-975-65.4-c1-0-29
Degree $2$
Conductor $975$
Sign $-0.465 + 0.885i$
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  + (0.793 − 1.37i)2-s + (−0.866 − 0.5i)3-s + (−0.259 − 0.450i)4-s + (−1.37 + 0.793i)6-s + (0.534 + 0.925i)7-s + 2.34·8-s + (0.499 + 0.866i)9-s + (−5.56 − 3.21i)11-s + 0.519i·12-s + (3.18 − 1.69i)13-s + 1.69·14-s + (2.38 − 4.13i)16-s + (−0.493 + 0.284i)17-s + 1.58·18-s + (3.65 − 2.11i)19-s + ⋯
L(s)  = 1  + (0.561 − 0.972i)2-s + (−0.499 − 0.288i)3-s + (−0.129 − 0.225i)4-s + (−0.561 + 0.324i)6-s + (0.202 + 0.349i)7-s + 0.830·8-s + (0.166 + 0.288i)9-s + (−1.67 − 0.969i)11-s + 0.150i·12-s + (0.882 − 0.471i)13-s + 0.453·14-s + (0.596 − 1.03i)16-s + (−0.119 + 0.0690i)17-s + 0.374·18-s + (0.839 − 0.484i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.465 + 0.885i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.465 + 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.465 + 0.885i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.465 + 0.885i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.975637 - 1.61479i\)
\(L(\frac12)\) \(\approx\) \(0.975637 - 1.61479i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.18 + 1.69i)T \)
good2 \( 1 + (-0.793 + 1.37i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (-0.534 - 0.925i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (5.56 + 3.21i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.493 - 0.284i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.65 + 2.11i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.03 - 1.17i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.64 + 6.31i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.15iT - 31T^{2} \)
37 \( 1 + (-4.68 + 8.11i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (6.49 + 3.74i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.53 + 3.19i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.76T + 47T^{2} \)
53 \( 1 - 9.77iT - 53T^{2} \)
59 \( 1 + (11.8 - 6.83i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.39 - 4.15i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.37 - 9.30i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (8.98 - 5.18i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 1.44T + 73T^{2} \)
79 \( 1 + 8.45T + 79T^{2} \)
83 \( 1 - 13.5T + 83T^{2} \)
89 \( 1 + (-10.9 - 6.31i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.35 - 2.34i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.26261703207543562748266326685, −8.916051357699110333186660799722, −7.949539972965507145582408326722, −7.37048231617152919675044422863, −5.87733994003495897227528252440, −5.40666618188129672929441258880, −4.30972399284652583989124601355, −3.11430144571918014670203885713, −2.38424549066840957403135182814, −0.836801298654725625107474957804, 1.51286215486111087952585599966, 3.24063539148499519812726233713, 4.78566480175904119767672023864, 4.83220190075939870771037629008, 5.99096619394943541668246978959, 6.76664176351494984067254573322, 7.57117979806329561501017589985, 8.260066165819587547211381670618, 9.564666165791756037622239754589, 10.47072713013824177751586123768

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