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Properties

Label 2-975-975.146-c0-0-1
Degree $2$
Conductor $975$
Sign $-0.666 - 0.745i$
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  + (0.336 + 1.58i)2-s + (0.994 + 0.104i)3-s + (−1.47 + 0.658i)4-s + (0.951 + 0.309i)5-s + (0.169 + 1.60i)6-s + (−0.5 + 0.866i)7-s + (−0.587 − 0.809i)8-s + (0.978 + 0.207i)9-s + (−0.169 + 1.60i)10-s + (−0.336 − 1.58i)11-s + (−1.53 + 0.500i)12-s + (−0.309 − 0.951i)13-s + (−1.53 − 0.5i)14-s + (0.913 + 0.406i)15-s + (−0.994 + 0.104i)17-s + 1.61i·18-s + ⋯
L(s)  = 1  + (0.336 + 1.58i)2-s + (0.994 + 0.104i)3-s + (−1.47 + 0.658i)4-s + (0.951 + 0.309i)5-s + (0.169 + 1.60i)6-s + (−0.5 + 0.866i)7-s + (−0.587 − 0.809i)8-s + (0.978 + 0.207i)9-s + (−0.169 + 1.60i)10-s + (−0.336 − 1.58i)11-s + (−1.53 + 0.500i)12-s + (−0.309 − 0.951i)13-s + (−1.53 − 0.5i)14-s + (0.913 + 0.406i)15-s + (−0.994 + 0.104i)17-s + 1.61i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.666 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.666 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.666 - 0.745i$
Analytic conductor: \(0.486588\)
Root analytic conductor: \(0.697558\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :0),\ -0.666 - 0.745i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.656852863\)
\(L(\frac12)\) \(\approx\) \(1.656852863\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.994 - 0.104i)T \)
5 \( 1 + (-0.951 - 0.309i)T \)
13 \( 1 + (0.309 + 0.951i)T \)
good2 \( 1 + (-0.336 - 1.58i)T + (-0.913 + 0.406i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.336 + 1.58i)T + (-0.913 + 0.406i)T^{2} \)
17 \( 1 + (0.994 - 0.104i)T + (0.978 - 0.207i)T^{2} \)
19 \( 1 + (0.104 + 0.994i)T + (-0.978 + 0.207i)T^{2} \)
23 \( 1 + (0.207 + 0.978i)T + (-0.913 + 0.406i)T^{2} \)
29 \( 1 + (0.978 + 0.207i)T^{2} \)
31 \( 1 + (0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.669 - 0.743i)T + (-0.104 - 0.994i)T^{2} \)
41 \( 1 + (0.743 + 0.669i)T + (0.104 + 0.994i)T^{2} \)
43 \( 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (-0.207 + 0.978i)T + (-0.913 - 0.406i)T^{2} \)
61 \( 1 + (-0.669 - 0.743i)T + (-0.104 + 0.994i)T^{2} \)
67 \( 1 + (-0.564 - 0.251i)T + (0.669 + 0.743i)T^{2} \)
71 \( 1 + (-0.669 + 0.743i)T^{2} \)
73 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
79 \( 1 + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (0.587 + 0.809i)T + (-0.309 + 0.951i)T^{2} \)
89 \( 1 + (-0.207 - 0.978i)T + (-0.913 + 0.406i)T^{2} \)
97 \( 1 + (0.564 - 0.251i)T + (0.669 - 0.743i)T^{2} \)
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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

−10.24083971562233189306654150775, −9.188548776385686445326775046667, −8.671602149017333583093833424434, −8.122209963684067073234488020542, −6.94814193066719998476953412345, −6.31621905913978760092540500257, −5.56347908955573443741848579849, −4.70741275984428327495533335842, −3.21448550399010397444616807544, −2.46350405951624965530189600947, 1.78936994083780491174032500563, 2.06338271119473914449267392653, 3.42638786742641969697655429748, 4.25846038384539198997712440724, 4.98848268394892767717407180956, 6.68565207870942656552929256243, 7.33243093796227450437020024312, 8.684738180416670513170202957351, 9.574503350740732757888230910108, 9.903024342637376749633590143100

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