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Properties

Label 2-975-195.59-c1-0-76
Degree $2$
Conductor $975$
Sign $-0.285 + 0.958i$
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  + (2.31 − 0.619i)2-s + (0.529 − 1.64i)3-s + (3.23 − 1.86i)4-s + (0.202 − 4.14i)6-s + (0.366 − 1.36i)7-s + (2.93 − 2.93i)8-s + (−2.43 − 1.74i)9-s + (1.69 − 0.453i)11-s + (−1.36 − 6.31i)12-s + (−3.23 + 1.59i)13-s − 3.38i·14-s + (1.23 − 2.13i)16-s + (1.85 − 1.07i)17-s + (−6.72 − 2.52i)18-s + (0.267 − i)19-s + ⋯
L(s)  = 1  + (1.63 − 0.438i)2-s + (0.305 − 0.952i)3-s + (1.61 − 0.933i)4-s + (0.0826 − 1.69i)6-s + (0.138 − 0.516i)7-s + (1.03 − 1.03i)8-s + (−0.813 − 0.582i)9-s + (0.510 − 0.136i)11-s + (−0.394 − 1.82i)12-s + (−0.896 + 0.443i)13-s − 0.904i·14-s + (0.308 − 0.533i)16-s + (0.450 − 0.260i)17-s + (−1.58 − 0.595i)18-s + (0.0614 − 0.229i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.285 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.285 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.285 + 0.958i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.285 + 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.58929 - 3.47237i\)
\(L(\frac12)\) \(\approx\) \(2.58929 - 3.47237i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.529 + 1.64i)T \)
5 \( 1 \)
13 \( 1 + (3.23 - 1.59i)T \)
good2 \( 1 + (-2.31 + 0.619i)T + (1.73 - i)T^{2} \)
7 \( 1 + (-0.366 + 1.36i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-1.69 + 0.453i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-1.85 + 1.07i)T + (8.5 - 14.7i)T^{2} \)
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 \( 1 + (-6.59 + 1.76i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.166 - 0.619i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (4.09 + 7.09i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.77 - 6.77i)T - 47iT^{2} \)
53 \( 1 + 4.62T + 53T^{2} \)
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 \( 1 + (2.26 + 8.46i)T + (-58.0 + 33.5i)T^{2} \)
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 \( 1 + (-12.5 - 3.36i)T + (84.0 + 48.5i)T^{2} \)
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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

−9.967078672281214976005292885151, −8.884880385480531900255275449584, −7.77807839236166717874807526443, −6.88888612954315926835254027724, −6.34567350579504562851379459813, −5.24268351173316550161890449506, −4.40011777728135281921843948039, −3.32041253208209880174164393647, −2.49106996900643194439132548937, −1.27736844747830377969414392069, 2.41690255219388201823540912368, 3.22541465383945900030842462154, 4.24636372225309073508819102254, 4.88363538597695817048612770664, 5.70078965632645154139863806642, 6.45880374020131466380853658411, 7.65805939088432825300938413960, 8.400366577589825859243911546709, 9.606391089713502116544239612691, 10.18009634304958185738855927833

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