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Properties

Label 2-975-13.10-c1-0-36
Degree $2$
Conductor $975$
Sign $-0.964 + 0.265i$
Analytic cond. $7.78541$
Root an. cond. $2.79023$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)3-s + (−1 + 1.73i)4-s + (1.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s + (−3 + 1.73i)11-s + 1.99·12-s + (−3.5 − 0.866i)13-s + (−1.99 − 3.46i)16-s + (3 + 1.73i)19-s − 1.73i·21-s + (−3 − 5.19i)23-s + 0.999·27-s + (−3 + 1.73i)28-s + (−3 − 5.19i)29-s − 1.73i·31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.5 + 0.866i)4-s + (0.566 + 0.327i)7-s + (−0.166 + 0.288i)9-s + (−0.904 + 0.522i)11-s + 0.577·12-s + (−0.970 − 0.240i)13-s + (−0.499 − 0.866i)16-s + (0.688 + 0.397i)19-s − 0.377i·21-s + (−0.625 − 1.08i)23-s + 0.192·27-s + (−0.566 + 0.327i)28-s + (−0.557 − 0.964i)29-s − 0.311i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.964 + 0.265i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.964 + 0.265i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.5 + 0.866i)T \)
good2 \( 1 + (1 - 1.73i)T^{2} \)
7 \( 1 + (-1.5 - 0.866i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (3 - 1.73i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3 - 1.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.73iT - 31T^{2} \)
37 \( 1 + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (6 - 3.46i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.46iT - 47T^{2} \)
53 \( 1 + 12T + 53T^{2} \)
59 \( 1 + (3 + 1.73i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.5 - 4.33i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (9 + 5.19i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 1.73iT - 73T^{2} \)
79 \( 1 + 11T + 79T^{2} \)
83 \( 1 + 13.8iT - 83T^{2} \)
89 \( 1 + (-6 + 3.46i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.5 - 2.59i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.651715615506368687536095045272, −8.584250861254240999058392445136, −7.75599880232631505058479938482, −7.50064747390494244364191570336, −6.18808933732839416372932720253, −5.08615404826134296847720932980, −4.49952413052877564035388332568, −3.04759202969689392374417843759, −2.06555189694570825403784157666, 0, 1.62653516659400084362747731031, 3.21632049637395201077392991657, 4.48956763572689170067062061150, 5.17307383378504087078451922997, 5.75750726649752517900508221771, 7.02286014279295953272144916573, 7.911030600551024386461923548373, 8.939055122074977073629508138444, 9.670613633660325487730464257980

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