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Properties

Label 2-975-195.59-c1-0-41
Degree $2$
Conductor $975$
Sign $-0.717 + 0.696i$
Analytic cond. $7.78541$
Root an. cond. $2.79023$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 − 0.866i)3-s + (−1.73 + i)4-s + (−1.13 + 4.23i)7-s + (1.5 + 2.59i)9-s + 3.46·12-s + (−2.5 + 2.59i)13-s + (1.99 − 3.46i)16-s + (0.830 − 3.09i)19-s + (5.36 − 5.36i)21-s − 5.19i·27-s + (−2.26 − 8.46i)28-s + (0.830 + 0.830i)31-s + (−5.19 − 3i)36-s + (−11.5 + 3.09i)37-s + (6 − 1.73i)39-s + ⋯
L(s)  = 1  + (−0.866 − 0.499i)3-s + (−0.866 + 0.5i)4-s + (−0.428 + 1.59i)7-s + (0.5 + 0.866i)9-s + 0.999·12-s + (−0.693 + 0.720i)13-s + (0.499 − 0.866i)16-s + (0.190 − 0.710i)19-s + (1.17 − 1.17i)21-s − 0.999i·27-s + (−0.428 − 1.59i)28-s + (0.149 + 0.149i)31-s + (−0.866 − 0.5i)36-s + (−1.90 + 0.509i)37-s + (0.960 − 0.277i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.717 + 0.696i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.717 + 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.717 + 0.696i$
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: \(1\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.717 + 0.696i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 + 0.866i)T \)
5 \( 1 \)
13 \( 1 + (2.5 - 2.59i)T \)
good2 \( 1 + (1.73 - i)T^{2} \)
7 \( 1 + (1.13 - 4.23i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.830 + 3.09i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.830 - 0.830i)T + 31iT^{2} \)
37 \( 1 + (11.5 - 3.09i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (0.866 + 1.5i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 47iT^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (4.33 + 7.5i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.23 + 15.7i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (7.63 + 7.63i)T + 73iT^{2} \)
79 \( 1 - 12.1T + 79T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-9.59 - 2.57i)T + (84.0 + 48.5i)T^{2} \)
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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

−9.478585298628673189594715102690, −8.970212705019022513513196587582, −8.066709382576667877114617059237, −7.07163707538560960904196913406, −6.22917228810694728175871876128, −5.22370059631119578225941289393, −4.72730066525408561025586538179, −3.21831308631525323536004267199, −2.02600426337840850800744741698, 0, 1.12532294035932675949907660779, 3.48621627604294440857766780420, 4.20201955865970357586715738415, 5.06039847031468960554823780266, 5.85852838761185637344588086488, 6.87873439185011293799539226271, 7.67933399907849720844594040389, 8.888409504240743323960306082402, 9.870503644005122717667776446497

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