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Properties

Label 2-975-195.164-c1-0-54
Degree $2$
Conductor $975$
Sign $0.431 + 0.902i$
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  + (0.707 + 0.707i)2-s + (−1.41 + i)3-s − 0.999i·4-s + (−1.70 − 0.292i)6-s + (1 + i)7-s + (2.12 − 2.12i)8-s + (1.00 − 2.82i)9-s + (−2.82 − 2.82i)11-s + (0.999 + 1.41i)12-s + (−3 + 2i)13-s + 1.41i·14-s + 1.00·16-s + (2.70 − 1.29i)18-s + (−1 − i)19-s + (−2.41 − 0.414i)21-s − 4.00i·22-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.816 + 0.577i)3-s − 0.499i·4-s + (−0.696 − 0.119i)6-s + (0.377 + 0.377i)7-s + (0.750 − 0.750i)8-s + (0.333 − 0.942i)9-s + (−0.852 − 0.852i)11-s + (0.288 + 0.408i)12-s + (−0.832 + 0.554i)13-s + 0.377i·14-s + 0.250·16-s + (0.638 − 0.304i)18-s + (−0.229 − 0.229i)19-s + (−0.526 − 0.0903i)21-s − 0.852i·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.431 + 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.431 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.431 + 0.902i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (749, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.431 + 0.902i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.926363 - 0.583990i\)
\(L(\frac12)\) \(\approx\) \(0.926363 - 0.583990i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.41 - i)T \)
5 \( 1 \)
13 \( 1 + (3 - 2i)T \)
good2 \( 1 + (-0.707 - 0.707i)T + 2iT^{2} \)
7 \( 1 + (-1 - i)T + 7iT^{2} \)
11 \( 1 + (2.82 + 2.82i)T + 11iT^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + (1 + i)T + 19iT^{2} \)
23 \( 1 + 8.48iT - 23T^{2} \)
29 \( 1 + 2.82iT - 29T^{2} \)
31 \( 1 + (5 + 5i)T + 31iT^{2} \)
37 \( 1 + (-1 - i)T + 37iT^{2} \)
41 \( 1 + (1.41 - 1.41i)T - 41iT^{2} \)
43 \( 1 - 6T + 43T^{2} \)
47 \( 1 + (-2.82 + 2.82i)T - 47iT^{2} \)
53 \( 1 + 5.65T + 53T^{2} \)
59 \( 1 + (-2.82 - 2.82i)T + 59iT^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 + (-5 + 5i)T - 67iT^{2} \)
71 \( 1 + (-2.82 + 2.82i)T - 71iT^{2} \)
73 \( 1 + (1 + i)T + 73iT^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 + (-5.65 - 5.65i)T + 83iT^{2} \)
89 \( 1 + (9.89 + 9.89i)T + 89iT^{2} \)
97 \( 1 + (7 - 7i)T - 97iT^{2} \)
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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

−9.976546583700732469681323799229, −9.187807993918501248828789661092, −8.119562550239981904369737622998, −6.98282190945396367693944130606, −6.22294262501307378293025083095, −5.44884629818054630039693489071, −4.82517950540750976713319566807, −3.99487938160348717048317674803, −2.34984455263533978237284609177, −0.46789272395221062628868678043, 1.62009956464111801742451288000, 2.66014765122719964393185980927, 3.97187030805399446596375311188, 5.03012589607614929023375351501, 5.45244063082564313508315268718, 7.01361111739257645236183053699, 7.54183144396936730020705590281, 8.107007983117112342326641087502, 9.550937014058362794384691283572, 10.55080320099149433765704856204

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