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Properties

Label 2-975-39.2-c1-0-17
Degree $2$
Conductor $975$
Sign $-0.872 + 0.487i$
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  + (−0.335 + 1.25i)2-s + (−0.518 + 1.65i)3-s + (0.275 + 0.159i)4-s + (−1.89 − 1.20i)6-s + (2.20 − 0.589i)7-s + (−2.12 + 2.12i)8-s + (−2.46 − 1.71i)9-s + (2.74 + 0.735i)11-s + (−0.406 + 0.373i)12-s + (−2.40 + 2.68i)13-s + 2.95i·14-s + (−1.63 − 2.82i)16-s + (−2.27 + 3.93i)17-s + (2.97 − 2.51i)18-s + (−0.775 − 2.89i)19-s + ⋯
L(s)  = 1  + (−0.237 + 0.885i)2-s + (−0.299 + 0.954i)3-s + (0.137 + 0.0796i)4-s + (−0.774 − 0.491i)6-s + (0.832 − 0.222i)7-s + (−0.751 + 0.751i)8-s + (−0.821 − 0.570i)9-s + (0.827 + 0.221i)11-s + (−0.117 + 0.107i)12-s + (−0.667 + 0.744i)13-s + 0.789i·14-s + (−0.407 − 0.706i)16-s + (−0.550 + 0.953i)17-s + (0.700 − 0.591i)18-s + (−0.177 − 0.664i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 + 0.487i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.872 + 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.872 + 0.487i$
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: \((2,\ 975,\ (\ :1/2),\ -0.872 + 0.487i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.280476 - 1.07648i\)
\(L(\frac12)\) \(\approx\) \(0.280476 - 1.07648i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.518 - 1.65i)T \)
5 \( 1 \)
13 \( 1 + (2.40 - 2.68i)T \)
good2 \( 1 + (0.335 - 1.25i)T + (-1.73 - i)T^{2} \)
7 \( 1 + (-2.20 + 0.589i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-2.74 - 0.735i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (2.27 - 3.93i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.775 + 2.89i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.223 + 0.387i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.98 - 2.87i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.62 - 4.62i)T - 31iT^{2} \)
37 \( 1 + (2.75 - 10.2i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (0.217 - 0.811i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-4.12 - 2.38i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.06 + 3.06i)T - 47iT^{2} \)
53 \( 1 + 8.28iT - 53T^{2} \)
59 \( 1 + (-1.67 - 6.25i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-5.66 + 9.80i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.96 + 2.40i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-12.8 + 3.43i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-1.56 - 1.56i)T + 73iT^{2} \)
79 \( 1 - 9.74T + 79T^{2} \)
83 \( 1 + (0.856 + 0.856i)T + 83iT^{2} \)
89 \( 1 + (-1.64 - 0.440i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (2.62 + 9.80i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.52307418755781320277610040785, −9.385797257903022074013762524649, −8.838307011462765218326251513278, −8.042446321288348771070896308896, −6.94757037933368508195918472235, −6.41909815972779957810117345535, −5.26459927065633999079017811448, −4.56299308414251970335156243545, −3.50411703465575273507230529354, −1.97620068814918760494114624716, 0.54837118048422533334171508971, 1.82414080707784023685516357622, 2.53632884200157459949472781208, 3.89721541146948675982039603348, 5.35920842301615272452368167815, 6.02840779257936422156739413486, 7.12736133311447207300312100123, 7.72853160479138919521008574477, 8.855976908292474026509434113680, 9.538583228241687067856615677535

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