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Properties

Label 2-975-975.191-c0-0-1
Degree $2$
Conductor $975$
Sign $-0.943 + 0.332i$
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  + (−0.251 − 0.564i)2-s + (0.207 − 0.978i)3-s + (0.413 − 0.459i)4-s + (−0.587 + 0.809i)5-s + (−0.604 + 0.128i)6-s + (−0.5 − 0.866i)7-s + (−0.951 − 0.309i)8-s + (−0.913 − 0.406i)9-s + (0.604 + 0.128i)10-s + (0.251 + 0.564i)11-s + (−0.363 − 0.500i)12-s + (0.809 − 0.587i)13-s + (−0.363 + 0.5i)14-s + (0.669 + 0.743i)15-s + (−0.207 − 0.978i)17-s + 0.618i·18-s + ⋯
L(s)  = 1  + (−0.251 − 0.564i)2-s + (0.207 − 0.978i)3-s + (0.413 − 0.459i)4-s + (−0.587 + 0.809i)5-s + (−0.604 + 0.128i)6-s + (−0.5 − 0.866i)7-s + (−0.951 − 0.309i)8-s + (−0.913 − 0.406i)9-s + (0.604 + 0.128i)10-s + (0.251 + 0.564i)11-s + (−0.363 − 0.500i)12-s + (0.809 − 0.587i)13-s + (−0.363 + 0.5i)14-s + (0.669 + 0.743i)15-s + (−0.207 − 0.978i)17-s + 0.618i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 + 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 + 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.943 + 0.332i$
Analytic conductor: \(0.486588\)
Root analytic conductor: \(0.697558\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :0),\ -0.943 + 0.332i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7844910157\)
\(L(\frac12)\) \(\approx\) \(0.7844910157\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.207 + 0.978i)T \)
5 \( 1 + (0.587 - 0.809i)T \)
13 \( 1 + (-0.809 + 0.587i)T \)
good2 \( 1 + (0.251 + 0.564i)T + (-0.669 + 0.743i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.251 - 0.564i)T + (-0.669 + 0.743i)T^{2} \)
17 \( 1 + (0.207 + 0.978i)T + (-0.913 + 0.406i)T^{2} \)
19 \( 1 + (0.978 - 0.207i)T + (0.913 - 0.406i)T^{2} \)
23 \( 1 + (0.406 + 0.913i)T + (-0.669 + 0.743i)T^{2} \)
29 \( 1 + (-0.913 - 0.406i)T^{2} \)
31 \( 1 + (-0.809 - 0.587i)T^{2} \)
37 \( 1 + (-0.104 - 0.994i)T + (-0.978 + 0.207i)T^{2} \)
41 \( 1 + (-0.994 + 0.104i)T + (0.978 - 0.207i)T^{2} \)
43 \( 1 + (-0.809 - 1.40i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.406 + 0.913i)T + (-0.669 - 0.743i)T^{2} \)
61 \( 1 + (0.104 - 0.994i)T + (-0.978 - 0.207i)T^{2} \)
67 \( 1 + (1.08 + 1.20i)T + (-0.104 + 0.994i)T^{2} \)
71 \( 1 + (0.104 + 0.994i)T^{2} \)
73 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (-0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.951 + 0.309i)T + (0.809 + 0.587i)T^{2} \)
89 \( 1 + (-0.406 - 0.913i)T + (-0.669 + 0.743i)T^{2} \)
97 \( 1 + (-1.08 + 1.20i)T + (-0.104 - 0.994i)T^{2} \)
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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

−10.09907407730014913519378227822, −9.069074291870934160609035036857, −8.050144267686844568044041141087, −7.17671030889853652791174909802, −6.60990172781766522506778373710, −5.95387989751729011334127145671, −4.17459676000513342035857318275, −3.10819767468947019993254109937, −2.28420086772296671246306701482, −0.77279085316847595321165205503, 2.34123269494058242456568465982, 3.64624413130197759045229735645, 4.16697920058513164205378931576, 5.74176194468989210122297078265, 6.02870737380203440423285315919, 7.40884524532634466958871710220, 8.374354746870385045982725337037, 8.919643646920828908239974939347, 9.189715110245934912105592950732, 10.65733265960111368930100550141

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