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Properties

Label 2-975-39.20-c1-0-19
Degree $2$
Conductor $975$
Sign $0.968 + 0.248i$
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  + (−0.619 − 2.31i)2-s + (1.28 + 1.16i)3-s + (−3.23 + 1.86i)4-s + (1.89 − 3.68i)6-s + (1.36 + 0.366i)7-s + (2.93 + 2.93i)8-s + (0.292 + 2.98i)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 + (6.72 − 2.52i)18-s + (−0.267 + i)19-s + ⋯
L(s)  = 1  + (−0.438 − 1.63i)2-s + (0.740 + 0.671i)3-s + (−1.61 + 0.933i)4-s + (0.773 − 1.50i)6-s + (0.516 + 0.138i)7-s + (1.03 + 1.03i)8-s + (0.0975 + 0.995i)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 + (1.58 − 0.595i)18-s + (−0.0614 + 0.229i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.968 + 0.248i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.968 + 0.248i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40481 - 0.177624i\)
\(L(\frac12)\) \(\approx\) \(1.40481 - 0.177624i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.28 - 1.16i)T \)
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 \( 1 + (3.36 - 12.5i)T + (-84.0 - 48.5i)T^{2} \)
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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

−9.978484470813034736432120507490, −9.421582461327772825422578892813, −8.500706712158626449662614738111, −8.155643363407482388213877013151, −6.72260246153510370187815382039, −5.00906042134597827310133254622, −4.38469223864612191127591637993, −3.33455561297016989540699136435, −2.50154054714759866632951481266, −1.48409773809990693716782208314, 0.74887867469646524759486648359, 2.48194853904285856975545779476, 3.95322874055895845960373750190, 5.16363788852156752157834662895, 6.05359613387373463467932531030, 6.78952669995070169648574272283, 7.70178442291182197156270806746, 8.171917195062399132980739346142, 8.699301030543650404603541031249, 9.659617281250028575303104375784

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