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Properties

Label 2-975-39.5-c1-0-48
Degree $2$
Conductor $975$
Sign $0.881 - 0.471i$
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  + (1.22 + 1.22i)2-s + (−1.22 + 1.22i)3-s + 0.999i·4-s − 2.99·6-s + (−2.22 − 2.22i)7-s + (1.22 − 1.22i)8-s − 2.99i·9-s + (1.77 − 1.77i)11-s + (−1.22 − 1.22i)12-s + (3 + 2i)13-s − 5.44i·14-s + 5·16-s − 3·17-s + (3.67 − 3.67i)18-s + (−0.449 + 0.449i)19-s + ⋯
L(s)  = 1  + (0.866 + 0.866i)2-s + (−0.707 + 0.707i)3-s + 0.499i·4-s − 1.22·6-s + (−0.840 − 0.840i)7-s + (0.433 − 0.433i)8-s − 0.999i·9-s + (0.535 − 0.535i)11-s + (−0.353 − 0.353i)12-s + (0.832 + 0.554i)13-s − 1.45i·14-s + 1.25·16-s − 0.727·17-s + (0.866 − 0.866i)18-s + (−0.103 + 0.103i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.881 - 0.471i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (551, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.881 - 0.471i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86313 + 0.467208i\)
\(L(\frac12)\) \(\approx\) \(1.86313 + 0.467208i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.22 - 1.22i)T \)
5 \( 1 \)
13 \( 1 + (-3 - 2i)T \)
good2 \( 1 + (-1.22 - 1.22i)T + 2iT^{2} \)
7 \( 1 + (2.22 + 2.22i)T + 7iT^{2} \)
11 \( 1 + (-1.77 + 1.77i)T - 11iT^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 + (0.449 - 0.449i)T - 19iT^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 + 7.89iT - 29T^{2} \)
31 \( 1 + (-7.12 + 7.12i)T - 31iT^{2} \)
37 \( 1 + (-0.449 - 0.449i)T + 37iT^{2} \)
41 \( 1 + (0.550 + 0.550i)T + 41iT^{2} \)
43 \( 1 - 10.8iT - 43T^{2} \)
47 \( 1 + (4.22 - 4.22i)T - 47iT^{2} \)
53 \( 1 + 7.89iT - 53T^{2} \)
59 \( 1 + (-0.674 + 0.674i)T - 59iT^{2} \)
61 \( 1 + 6.79T + 61T^{2} \)
67 \( 1 + (-2.67 + 2.67i)T - 67iT^{2} \)
71 \( 1 + (-6 - 6i)T + 71iT^{2} \)
73 \( 1 + (1.55 + 1.55i)T + 73iT^{2} \)
79 \( 1 - 8.44T + 79T^{2} \)
83 \( 1 + (4.22 + 4.22i)T + 83iT^{2} \)
89 \( 1 + (5.44 - 5.44i)T - 89iT^{2} \)
97 \( 1 + (-1.55 + 1.55i)T - 97iT^{2} \)
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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

−9.961476341849722094416409028336, −9.489016622171036789303566390203, −8.296745055477894985428636890802, −7.03877124000038589868151527787, −6.31443118979332452524790642037, −6.06331082255423531009545226394, −4.70053725934192124731554610787, −4.15705852031233149477988712729, −3.32829053392045572107922168683, −0.850429862198230400519950295759, 1.35072346263529726640775917642, 2.55587199777379729822290132080, 3.42858152299988746871570160196, 4.72813144458092804545313944023, 5.43845098753456820845569809101, 6.43407486457444916672045961260, 7.08220434841651380894382335065, 8.370258425795636018554845877560, 9.107287947254217623630714823577, 10.46058308438521363881010382001

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