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Properties

Label 2-975-195.128-c0-0-1
Degree $2$
Conductor $975$
Sign $-0.806 + 0.591i$
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  + (0.258 − 0.965i)3-s + (−0.5 + 0.866i)4-s + (−1.67 − 0.965i)7-s + (−0.866 − 0.499i)9-s + (0.707 + 0.707i)12-s + (−0.707 − 0.707i)13-s + (−0.499 − 0.866i)16-s + (−0.133 − 0.5i)19-s + (−1.36 + 1.36i)21-s + (−0.707 + 0.707i)27-s + (1.67 − 0.965i)28-s + (−1 − i)31-s + (0.866 − 0.5i)36-s + (1.22 − 0.707i)37-s + (−0.866 + 0.500i)39-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)3-s + (−0.5 + 0.866i)4-s + (−1.67 − 0.965i)7-s + (−0.866 − 0.499i)9-s + (0.707 + 0.707i)12-s + (−0.707 − 0.707i)13-s + (−0.499 − 0.866i)16-s + (−0.133 − 0.5i)19-s + (−1.36 + 1.36i)21-s + (−0.707 + 0.707i)27-s + (1.67 − 0.965i)28-s + (−1 − i)31-s + (0.866 − 0.5i)36-s + (1.22 − 0.707i)37-s + (−0.866 + 0.500i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.806 + 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.806 + 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.806 + 0.591i$
Analytic conductor: \(0.486588\)
Root analytic conductor: \(0.697558\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (518, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :0),\ -0.806 + 0.591i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4770068764\)
\(L(\frac12)\) \(\approx\) \(0.4770068764\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 \)
13 \( 1 + (0.707 + 0.707i)T \)
good2 \( 1 + (0.5 - 0.866i)T^{2} \)
7 \( 1 + (1.67 + 0.965i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.866 - 0.5i)T^{2} \)
17 \( 1 + (-0.866 + 0.5i)T^{2} \)
19 \( 1 + (0.133 + 0.5i)T + (-0.866 + 0.5i)T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (1 + i)T + iT^{2} \)
37 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.866 + 0.5i)T^{2} \)
43 \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (-0.866 + 0.5i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.258 + 0.448i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.866 + 0.5i)T^{2} \)
73 \( 1 + 0.517T + T^{2} \)
79 \( 1 + 1.73iT - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.866 + 0.5i)T^{2} \)
97 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.558792961120227486003423492532, −9.179437800677244381939696161603, −7.924631725216075511215955430115, −7.44156031614745241473190203076, −6.72099146116220689393661593049, −5.78428790005799739314096356232, −4.29873082988888917873026029493, −3.34043120540219849526030771250, −2.62848415218573275370228507111, −0.41762239906836221844077613509, 2.28848271879952687304960766908, 3.37667731678366515524673427609, 4.38483705144429478310610878940, 5.41058843094633473550682567934, 6.02091992790756394536607424505, 6.96926710814499850980263195397, 8.522263832542543201189977401925, 9.122349737350101839610905319048, 9.733787965583042179024746977603, 10.13590195543860675777589134429

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