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Properties

Label 2-975-195.89-c1-0-58
Degree $2$
Conductor $975$
Sign $0.983 + 0.181i$
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  + (1.5 − 0.866i)3-s + (1.73 + i)4-s + (2.86 − 0.767i)7-s + (1.5 − 2.59i)9-s + 3.46·12-s + (2.5 + 2.59i)13-s + (1.99 + 3.46i)16-s + (−7.83 + 2.09i)19-s + (3.63 − 3.63i)21-s − 5.19i·27-s + (5.73 + 1.53i)28-s + (−7.83 − 7.83i)31-s + (5.19 − 3i)36-s + (−0.562 + 2.09i)37-s + (6 + 1.73i)39-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s + (0.866 + 0.5i)4-s + (1.08 − 0.290i)7-s + (0.5 − 0.866i)9-s + 0.999·12-s + (0.693 + 0.720i)13-s + (0.499 + 0.866i)16-s + (−1.79 + 0.481i)19-s + (0.792 − 0.792i)21-s − 0.999i·27-s + (1.08 + 0.290i)28-s + (−1.40 − 1.40i)31-s + (0.866 − 0.5i)36-s + (−0.0924 + 0.344i)37-s + (0.960 + 0.277i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.181i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 + 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.983 + 0.181i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (674, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.983 + 0.181i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.89546 - 0.264903i\)
\(L(\frac12)\) \(\approx\) \(2.89546 - 0.264903i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.86 + 0.767i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (7.83 - 2.09i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (7.83 + 7.83i)T + 31iT^{2} \)
37 \( 1 + (0.562 - 2.09i)T + (-32.0 - 18.5i)T^{2} \)
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 \( 1 + (-0.767 - 0.205i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-9.36 - 9.36i)T + 73iT^{2} \)
79 \( 1 + 12.1T + 79T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (4.40 + 16.4i)T + (-84.0 + 48.5i)T^{2} \)
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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

−9.979941490625738008903829836977, −8.787882646956911240042579853282, −8.264606771269844027594667926486, −7.57521813512348543887760948784, −6.76499403295368720976560821925, −5.98283489973279875015884412358, −4.33722521695960552316982550276, −3.64422048473657452406481746743, −2.25523319644310250178072513440, −1.62351836894839337549805214989, 1.60615906871848476247335009106, 2.45623508298647302971280871973, 3.62600757946305135886453604994, 4.80017184354271570003084961541, 5.59437877827847246052872478297, 6.71961744510472894203788798444, 7.65033577376203830367911617176, 8.461494785559430782098743698133, 9.019330021640099967046399544180, 10.25549565270358197778607220960

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