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Properties

Label 2-975-195.29-c0-0-1
Degree $2$
Conductor $975$
Sign $0.322 + 0.946i$
Analytic cond. $0.486588$
Root an. cond. $0.697558$
Motivic weight $0$
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 − 0.5i)3-s + (0.5 − 0.866i)4-s + (−0.866 − 0.5i)7-s + (0.499 − 0.866i)9-s − 0.999i·12-s + i·13-s + (−0.499 − 0.866i)16-s + (−0.5 + 0.866i)19-s − 0.999·21-s − 0.999i·27-s + (−0.866 + 0.499i)28-s + 2·31-s + (−0.499 − 0.866i)36-s + (−1.73 + i)37-s + (0.5 + 0.866i)39-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)3-s + (0.5 − 0.866i)4-s + (−0.866 − 0.5i)7-s + (0.499 − 0.866i)9-s − 0.999i·12-s + i·13-s + (−0.499 − 0.866i)16-s + (−0.5 + 0.866i)19-s − 0.999·21-s − 0.999i·27-s + (−0.866 + 0.499i)28-s + 2·31-s + (−0.499 − 0.866i)36-s + (−1.73 + i)37-s + (0.5 + 0.866i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.322 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.322 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.322 + 0.946i$
Analytic conductor: \(0.486588\)
Root analytic conductor: \(0.697558\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (224, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :0),\ 0.322 + 0.946i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.358490199\)
\(L(\frac12)\) \(\approx\) \(1.358490199\)
\(L(1)\) 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 \( 1 - iT \)
good2 \( 1 + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 - 2T + T^{2} \)
37 \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + iT - T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2} \)
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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

−10.00071826969149236424756583212, −9.316273731660792870832350777994, −8.415800432743590563584239387009, −7.38687004474828700368709590707, −6.58222945416412569508980364314, −6.18999004559644605345422925579, −4.66490929315396902704700856610, −3.57494138434202337835865687000, −2.47197013082709774899791691400, −1.34768790193108976191285941272, 2.34771797430569818960100333045, 3.02096976755714187910981716854, 3.84391005166167262508875013627, 5.00450107030999870098180458543, 6.26367363529449067205461166082, 7.13190662600404668670397688462, 8.029569941269200585931172766697, 8.670703045732724951133760637833, 9.409615506310209289163852114773, 10.34310638821380455449856953471

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