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Properties

Label 2-975-13.10-c1-0-15
Degree $2$
Conductor $975$
Sign $0.0250 - 0.999i$
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  + (−2.04 + 1.18i)2-s + (−0.5 − 0.866i)3-s + (1.78 − 3.09i)4-s + (2.04 + 1.18i)6-s + (3.93 + 2.27i)7-s + 3.71i·8-s + (−0.499 + 0.866i)9-s + (3.74 − 2.16i)11-s − 3.57·12-s + (−1.84 + 3.10i)13-s − 10.7·14-s + (−0.807 − 1.39i)16-s + (−2.76 + 4.78i)17-s − 2.36i·18-s + (3.41 + 1.97i)19-s + ⋯
L(s)  = 1  + (−1.44 + 0.834i)2-s + (−0.288 − 0.499i)3-s + (0.892 − 1.54i)4-s + (0.834 + 0.481i)6-s + (1.48 + 0.858i)7-s + 1.31i·8-s + (−0.166 + 0.288i)9-s + (1.12 − 0.652i)11-s − 1.03·12-s + (−0.510 + 0.859i)13-s − 2.86·14-s + (−0.201 − 0.349i)16-s + (−0.670 + 1.16i)17-s − 0.556i·18-s + (0.783 + 0.452i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0250 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0250 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.0250 - 0.999i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.0250 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.570217 + 0.556103i\)
\(L(\frac12)\) \(\approx\) \(0.570217 + 0.556103i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 \)
13 \( 1 + (1.84 - 3.10i)T \)
good2 \( 1 + (2.04 - 1.18i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (-3.93 - 2.27i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3.74 + 2.16i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.76 - 4.78i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.41 - 1.97i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.364 - 0.630i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.97 + 3.42i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.599iT - 31T^{2} \)
37 \( 1 + (-3.91 + 2.26i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (8.97 - 5.18i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.45 + 9.44i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.42iT - 47T^{2} \)
53 \( 1 + 0.805T + 53T^{2} \)
59 \( 1 + (-9.71 - 5.60i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.20 - 10.7i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.12 - 5.26i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.898 - 0.518i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 6.45iT - 73T^{2} \)
79 \( 1 - 4.22T + 79T^{2} \)
83 \( 1 + 7.69iT - 83T^{2} \)
89 \( 1 + (5.93 - 3.42i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.61 + 5.54i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.980577295500011978638556143545, −8.893408681197235200482972521773, −8.673200691923310395813912796165, −7.79852931219260706364788121122, −7.04847891062229474586439587323, −6.13515514355302536847183486426, −5.51230547340840635698253404933, −4.17836753742799468599519430867, −2.04915389597697743968645763642, −1.28269718778159259430593353547, 0.72049055190964684200508751094, 1.80302968320669973718034446768, 3.14161218009464573446493093682, 4.45207193438853277072928219616, 5.14572923607343959157831698090, 6.90269269690639264564041605803, 7.52675897730803404193116820711, 8.306286488102971876859199020290, 9.292764024717406332489020333637, 9.723121866046649097664166820679

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