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Properties

Label 2-975-195.44-c1-0-13
Degree $2$
Conductor $975$
Sign $-0.816 - 0.577i$
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 + 2.99i·6-s + (−0.224 + 0.224i)7-s + (−1.22 − 1.22i)8-s − 2.99i·9-s + (−4.22 + 4.22i)11-s + (−1.22 − 1.22i)12-s + (2 − 3i)13-s − 0.550i·14-s + 5·16-s + 3i·17-s + (3.67 + 3.67i)18-s + (−4.44 + 4.44i)19-s + ⋯
L(s)  = 1  + (−0.866 + 0.866i)2-s + (0.707 − 0.707i)3-s − 0.499i·4-s + 1.22i·6-s + (−0.0849 + 0.0849i)7-s + (−0.433 − 0.433i)8-s − 0.999i·9-s + (−1.27 + 1.27i)11-s + (−0.353 − 0.353i)12-s + (0.554 − 0.832i)13-s − 0.147i·14-s + 1.25·16-s + 0.727i·17-s + (0.866 + 0.866i)18-s + (−1.02 + 1.02i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.816 - 0.577i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (824, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.816 - 0.577i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.204941 + 0.644495i\)
\(L(\frac12)\) \(\approx\) \(0.204941 + 0.644495i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 + 3i)T \)
good2 \( 1 + (1.22 - 1.22i)T - 2iT^{2} \)
7 \( 1 + (0.224 - 0.224i)T - 7iT^{2} \)
11 \( 1 + (4.22 - 4.22i)T - 11iT^{2} \)
17 \( 1 - 3iT - 17T^{2} \)
19 \( 1 + (4.44 - 4.44i)T - 19iT^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 - 1.89iT - 29T^{2} \)
31 \( 1 + (5.12 - 5.12i)T - 31iT^{2} \)
37 \( 1 + (-4.44 + 4.44i)T - 37iT^{2} \)
41 \( 1 + (-5.44 - 5.44i)T + 41iT^{2} \)
43 \( 1 - 1.10T + 43T^{2} \)
47 \( 1 + (-1.77 - 1.77i)T + 47iT^{2} \)
53 \( 1 + 1.89T + 53T^{2} \)
59 \( 1 + (6.67 - 6.67i)T - 59iT^{2} \)
61 \( 1 - 12.7T + 61T^{2} \)
67 \( 1 + (4.67 + 4.67i)T + 67iT^{2} \)
71 \( 1 + (6 + 6i)T + 71iT^{2} \)
73 \( 1 + (6.44 - 6.44i)T - 73iT^{2} \)
79 \( 1 + 3.55T + 79T^{2} \)
83 \( 1 + (-1.77 + 1.77i)T - 83iT^{2} \)
89 \( 1 + (0.550 - 0.550i)T - 89iT^{2} \)
97 \( 1 + (-6.44 - 6.44i)T + 97iT^{2} \)
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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

−10.05040731608988017627993546741, −9.244451172538015909781944383267, −8.456779135525758360200979308033, −7.68382659516806603493750985031, −7.47029359387372984846668236904, −6.30902863981445129670784588405, −5.60364183395771062375815276307, −3.97396734067456704386158078298, −2.87336833384631559892765603574, −1.54588750520708813610928810593, 0.37064218910755721785480187854, 2.26034172298629930610112340900, 2.85120203282057248105762968192, 4.04241530940077300132299760967, 5.15294617301200059572870642820, 6.19458081271764468394518962066, 7.58022928789655768811419397837, 8.577144861453592322349874178642, 8.795595859616695962110111668960, 9.707490609606473463736516145514

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