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Properties

Label 2-975-39.2-c1-0-47
Degree $2$
Conductor $975$
Sign $0.302 - 0.953i$
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  + (0.866 + 1.5i)3-s + (1.73 + i)4-s + (4.23 − 1.13i)7-s + (−1.5 + 2.59i)9-s + 3.46i·12-s + (−2.59 + 2.5i)13-s + (1.99 + 3.46i)16-s + (−0.830 − 3.09i)19-s + (5.36 + 5.36i)21-s − 5.19·27-s + (8.46 + 2.26i)28-s + (0.830 − 0.830i)31-s + (−5.19 + 3i)36-s + (3.09 − 11.5i)37-s + (−6 − 1.73i)39-s + ⋯
L(s)  = 1  + (0.499 + 0.866i)3-s + (0.866 + 0.5i)4-s + (1.59 − 0.428i)7-s + (−0.5 + 0.866i)9-s + 0.999i·12-s + (−0.720 + 0.693i)13-s + (0.499 + 0.866i)16-s + (−0.190 − 0.710i)19-s + (1.17 + 1.17i)21-s − 1.00·27-s + (1.59 + 0.428i)28-s + (0.149 − 0.149i)31-s + (−0.866 + 0.5i)36-s + (0.509 − 1.90i)37-s + (−0.960 − 0.277i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.302 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.302 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.302 - 0.953i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (626, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.302 - 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.09160 + 1.53138i\)
\(L(\frac12)\) \(\approx\) \(2.09160 + 1.53138i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 - 1.5i)T \)
5 \( 1 \)
13 \( 1 + (2.59 - 2.5i)T \)
good2 \( 1 + (-1.73 - i)T^{2} \)
7 \( 1 + (-4.23 + 1.13i)T + (6.06 - 3.5i)T^{2} \)
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 \( 1 + (-3.09 + 11.5i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-1.5 - 0.866i)T + (21.5 + 37.2i)T^{2} \)
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 \( 1 + (15.7 + 4.23i)T + (58.0 + 33.5i)T^{2} \)
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 \( 1 + (-2.57 - 9.59i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.34568680075369608964063744368, −9.250338700671148978116029563317, −8.479596080485403830249476034865, −7.66034587358362139688645300985, −7.14552604719156323297839360948, −5.72957626854604558177339413545, −4.64709422296980249395175205425, −4.06620227419204619778280096180, −2.71659025428272463979998148386, −1.84571881612551019092567895965, 1.26723171417634862529441970980, 2.11345412132740137357206902414, 3.05433005084400286477855878564, 4.73625580593607963314510702454, 5.64294962458155539760536222082, 6.46435985444908508362497468530, 7.55916021418293719319279586238, 7.925094221368922244924076223092, 8.777855000012811156800368792087, 9.918421415159163612320951851350

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