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Properties

Label 2-975-65.4-c1-0-2
Degree $2$
Conductor $975$
Sign $-0.980 + 0.194i$
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  + (−0.366 + 0.633i)2-s + (−0.866 − 0.5i)3-s + (0.732 + 1.26i)4-s + (0.633 − 0.366i)6-s + (2.23 + 3.86i)7-s − 2.53·8-s + (0.499 + 0.866i)9-s + (−3 − 1.73i)11-s − 1.46i·12-s + (−3.5 + 0.866i)13-s − 3.26·14-s + (−0.535 + 0.928i)16-s + (−5.83 + 3.36i)17-s − 0.732·18-s + (4.73 − 2.73i)19-s + ⋯
L(s)  = 1  + (−0.258 + 0.448i)2-s + (−0.499 − 0.288i)3-s + (0.366 + 0.633i)4-s + (0.258 − 0.149i)6-s + (0.843 + 1.46i)7-s − 0.896·8-s + (0.166 + 0.288i)9-s + (−0.904 − 0.522i)11-s − 0.422i·12-s + (−0.970 + 0.240i)13-s − 0.873·14-s + (−0.133 + 0.232i)16-s + (−1.41 + 0.816i)17-s − 0.172·18-s + (1.08 − 0.626i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 + 0.194i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 + 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.980 + 0.194i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.980 + 0.194i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0610919 - 0.623193i\)
\(L(\frac12)\) \(\approx\) \(0.0610919 - 0.623193i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.5 - 0.866i)T \)
good2 \( 1 + (0.366 - 0.633i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (-2.23 - 3.86i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (3 + 1.73i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (5.83 - 3.36i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.73 + 2.73i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.464 + 0.267i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.36 - 2.36i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.19iT - 31T^{2} \)
37 \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.56 - 2.63i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.232 + 0.133i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 0.196T + 47T^{2} \)
53 \( 1 + 6.92iT - 53T^{2} \)
59 \( 1 + (6.29 - 3.63i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.23 + 3.86i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.23 - 10.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (11.0 - 6.36i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 15.3T + 73T^{2} \)
79 \( 1 + 1.92T + 79T^{2} \)
83 \( 1 - 2.53T + 83T^{2} \)
89 \( 1 + (-1.09 - 0.633i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.23 - 14.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.63469186719774628849733038691, −9.267250229349379839105782404258, −8.686519241186540461640521657965, −7.87017756495031945721219214846, −7.22577491817874276141009544835, −6.14310973732141578409462640590, −5.48822805131380004413032203107, −4.53785950985929038582079028864, −2.82968354386930019317707583891, −2.11469934058114748316153610013, 0.31248525208518421653984280236, 1.65485661416794654420197058996, 2.92109712901595911801998121848, 4.49240510380355209543938562531, 4.96778285728543956968355378161, 6.05727543774034306811299306411, 7.31917223791362240147429197302, 7.51619069191906197519944198834, 9.028817628347478865619542770186, 9.990140773213366307083602079165

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