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Properties

Label 2-975-65.9-c1-0-31
Degree $2$
Conductor $975$
Sign $0.997 - 0.0727i$
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  + (1.73 + i)2-s + (−0.866 − 0.5i)3-s + (0.999 + 1.73i)4-s + (−0.999 − 1.73i)6-s + (0.499 + 0.866i)9-s + (1 − 1.73i)11-s − 1.99i·12-s + (0.866 − 3.5i)13-s + (1.99 − 3.46i)16-s + (3.46 − 2i)17-s + 1.99i·18-s + (2.5 + 4.33i)19-s + (3.46 − 1.99i)22-s + (3.46 + 2i)23-s + (5 − 5.19i)26-s − 0.999i·27-s + ⋯
L(s)  = 1  + (1.22 + 0.707i)2-s + (−0.499 − 0.288i)3-s + (0.499 + 0.866i)4-s + (−0.408 − 0.707i)6-s + (0.166 + 0.288i)9-s + (0.301 − 0.522i)11-s − 0.577i·12-s + (0.240 − 0.970i)13-s + (0.499 − 0.866i)16-s + (0.840 − 0.485i)17-s + 0.471i·18-s + (0.573 + 0.993i)19-s + (0.738 − 0.426i)22-s + (0.722 + 0.417i)23-s + (0.980 − 1.01i)26-s − 0.192i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.997 - 0.0727i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (724, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.997 - 0.0727i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.70716 + 0.0985501i\)
\(L(\frac12)\) \(\approx\) \(2.70716 + 0.0985501i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 \)
13 \( 1 + (-0.866 + 3.5i)T \)
good2 \( 1 + (-1.73 - i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.46 + 2i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.46 - 2i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1 - 1.73i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - T + 31T^{2} \)
37 \( 1 + (5.19 + 3i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.92 + 4i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 4iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + (3 + 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.46 + 2i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3 - 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 15iT - 73T^{2} \)
79 \( 1 + T + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 + (7 - 12.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.79 - 4.5i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.15477575644664877389273691327, −9.144333718630169416974056171875, −7.87545444222444278390589494819, −7.33528642020846888023504212290, −6.32770463997124066939621662736, −5.59458731995373797297570016399, −5.13341868592944432229197099736, −3.83627660718733821800934852774, −3.06968291334659682126648328223, −1.06170457845702073262918400216, 1.48147675661962785616099188900, 2.80844135289196604487656652734, 3.85427221551153330167066264471, 4.61197991413883355996881412841, 5.34653652473008290400621277516, 6.31400745214850691623955266838, 7.16153346359525283730648590664, 8.446351038159070732495446218144, 9.423759951190691172405429085337, 10.23814491747742186784762040799

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