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Properties

Label 2-975-195.2-c0-0-1
Degree $2$
Conductor $975$
Sign $0.581 - 0.813i$
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  + (0.965 + 0.258i)3-s + (−0.5 + 0.866i)4-s + (0.448 + 0.258i)7-s + (0.866 + 0.499i)9-s + (−0.707 + 0.707i)12-s + (0.707 − 0.707i)13-s + (−0.499 − 0.866i)16-s + (−1.86 + 0.5i)19-s + (0.366 + 0.366i)21-s + (0.707 + 0.707i)27-s + (−0.448 + 0.258i)28-s + (−1 + i)31-s + (−0.866 + 0.5i)36-s + (1.22 − 0.707i)37-s + (0.866 − 0.500i)39-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)3-s + (−0.5 + 0.866i)4-s + (0.448 + 0.258i)7-s + (0.866 + 0.499i)9-s + (−0.707 + 0.707i)12-s + (0.707 − 0.707i)13-s + (−0.499 − 0.866i)16-s + (−1.86 + 0.5i)19-s + (0.366 + 0.366i)21-s + (0.707 + 0.707i)27-s + (−0.448 + 0.258i)28-s + (−1 + i)31-s + (−0.866 + 0.5i)36-s + (1.22 − 0.707i)37-s + (0.866 − 0.500i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.581 - 0.813i$
Analytic conductor: \(0.486588\)
Root analytic conductor: \(0.697558\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (782, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :0),\ 0.581 - 0.813i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.297818827\)
\(L(\frac12)\) \(\approx\) \(1.297818827\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.965 - 0.258i)T \)
5 \( 1 \)
13 \( 1 + (-0.707 + 0.707i)T \)
good2 \( 1 + (0.5 - 0.866i)T^{2} \)
7 \( 1 + (-0.448 - 0.258i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.866 + 0.5i)T^{2} \)
17 \( 1 + (0.866 - 0.5i)T^{2} \)
19 \( 1 + (1.86 - 0.5i)T + (0.866 - 0.5i)T^{2} \)
23 \( 1 + (0.866 + 0.5i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (1 - i)T - iT^{2} \)
37 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.866 - 0.5i)T^{2} \)
43 \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + (0.866 - 0.5i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.965 + 1.67i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.866 - 0.5i)T^{2} \)
73 \( 1 + 1.93T + T^{2} \)
79 \( 1 + 1.73iT - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.866 - 0.5i)T^{2} \)
97 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
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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

−10.32621041092565913572036518701, −9.132532457943673819635600563514, −8.697533038284139551428477309134, −8.018236302454931214208431847978, −7.34347863478285431898603783723, −6.04873856014415344863373053145, −4.78074159184194031656514276196, −3.99413508445160320111024157355, −3.15256344750021956163308256830, −2.00088615689049578887590151992, 1.37864044940757262956659666685, 2.43317492474050918499729699918, 4.07117782560229026840241731625, 4.46917894273611701970892850642, 5.92364715801154824103318348360, 6.65764210408173620413305416647, 7.70823522378414389032471415227, 8.634921690109889496992569836299, 9.090614799700010102289628757016, 9.953273338287148198122726981420

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