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Properties

Label 2-975-13.4-c1-0-24
Degree $2$
Conductor $975$
Sign $-0.360 - 0.932i$
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  + (2.23 + 1.29i)2-s + (−0.5 + 0.866i)3-s + (2.34 + 4.05i)4-s + (−2.23 + 1.29i)6-s + (3.35 − 1.93i)7-s + 6.93i·8-s + (−0.499 − 0.866i)9-s + (1.25 + 0.727i)11-s − 4.68·12-s + (2.77 + 2.29i)13-s + 10.0·14-s + (−4.28 + 7.42i)16-s + (−1.85 − 3.21i)17-s − 2.58i·18-s + (−4.01 + 2.31i)19-s + ⋯
L(s)  = 1  + (1.58 + 0.914i)2-s + (−0.288 + 0.499i)3-s + (1.17 + 2.02i)4-s + (−0.914 + 0.527i)6-s + (1.26 − 0.731i)7-s + 2.45i·8-s + (−0.166 − 0.288i)9-s + (0.379 + 0.219i)11-s − 1.35·12-s + (0.770 + 0.637i)13-s + 2.67·14-s + (−1.07 + 1.85i)16-s + (−0.450 − 0.780i)17-s − 0.609i·18-s + (−0.920 + 0.531i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.360 - 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.360 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.360 - 0.932i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.360 - 0.932i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.27339 + 3.31541i\)
\(L(\frac12)\) \(\approx\) \(2.27339 + 3.31541i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 \)
13 \( 1 + (-2.77 - 2.29i)T \)
good2 \( 1 + (-2.23 - 1.29i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (-3.35 + 1.93i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.25 - 0.727i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.85 + 3.21i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.01 - 2.31i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.79 - 4.83i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.944 + 1.63i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 8.50iT - 31T^{2} \)
37 \( 1 + (8.21 + 4.74i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.90 - 3.40i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.81 + 8.33i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 5.84iT - 47T^{2} \)
53 \( 1 + 12.9T + 53T^{2} \)
59 \( 1 + (-8.34 + 4.81i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.669 - 1.15i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.16 + 1.82i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.09 - 3.51i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 3.43iT - 73T^{2} \)
79 \( 1 - 15.4T + 79T^{2} \)
83 \( 1 - 0.637iT - 83T^{2} \)
89 \( 1 + (-2.83 - 1.63i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-12.8 + 7.39i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.60175988680664543465753614793, −9.305064967995678190373328006359, −8.203910626279758183361351829536, −7.51995925963193592588530141295, −6.62386553327289996120339755725, −5.84469286402714146309752661682, −4.91221434357099854874658418071, −4.21969456936271841011582178231, −3.69620687107748789273506399656, −1.97916566119735514936368497259, 1.40130977895668327985474277425, 2.21135698643471402945680444374, 3.37758438652781438644032059023, 4.53901123271998741121230226311, 5.14991599602591694931854171344, 6.10347527139368241519452776472, 6.64961125485057840781264006429, 8.202529831565344958971189157071, 8.775335713630845500425816224925, 10.48531625737648465384230804920

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