Properties

Label 2-975-195.2-c0-0-0
Degree $2$
Conductor $975$
Sign $-0.913 - 0.406i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.913 - 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.913 - 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.913 - 0.406i$
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: \((2,\ 975,\ (\ :0),\ -0.913 - 0.406i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2224220517\)
\(L(\frac12)\) \(\approx\) \(0.2224220517\)
\(L(1)\) 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

−10.55376403790792924731089337669, −9.853919137239872863053128596207, −8.848890163716550086136392129615, −8.039999090317525187930308162107, −6.97387185160634084362339120813, −6.58960122946672671106058247119, −5.26476153063917705151307982981, −4.45438172651870243624124860761, −3.56688904395784709297476869362, −2.01066125703874419260014822840, 0.22888063412094698291290875462, 2.06310941971067201955068874869, 3.78971912271402120139586310613, 4.77630670119967318378677245032, 5.48172168487979534572861317056, 6.26967277577634171569013439037, 7.00655344955167514311810134821, 8.331076592125628208659127566127, 9.330881987264714266549613145288, 9.869925839390100053556356908485

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