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Properties

Label 2-1575-7.5-c0-0-1
Degree $2$
Conductor $1575$
Sign $0.605 - 0.795i$
Analytic cond. $0.786027$
Root an. cond. $0.886581$
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  + (0.5 + 0.866i)4-s + (0.5 + 0.866i)7-s − 1.73i·13-s + (−0.499 + 0.866i)16-s + (1.5 + 0.866i)19-s + (−0.499 + 0.866i)28-s + (−0.5 + 0.866i)37-s − 2·43-s + (−0.499 + 0.866i)49-s + (1.49 − 0.866i)52-s + (−1.5 − 0.866i)61-s − 0.999·64-s + (0.5 + 0.866i)67-s + (1.5 − 0.866i)73-s + 1.73i·76-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)4-s + (0.5 + 0.866i)7-s − 1.73i·13-s + (−0.499 + 0.866i)16-s + (1.5 + 0.866i)19-s + (−0.499 + 0.866i)28-s + (−0.5 + 0.866i)37-s − 2·43-s + (−0.499 + 0.866i)49-s + (1.49 − 0.866i)52-s + (−1.5 − 0.866i)61-s − 0.999·64-s + (0.5 + 0.866i)67-s + (1.5 − 0.866i)73-s + 1.73i·76-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(0.786027\)
Root analytic conductor: \(0.886581\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :0),\ 0.605 - 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.322807237\)
\(L(\frac12)\) \(\approx\) \(1.322807237\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + 1.73iT - T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 2T + T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + 1.73iT - T^{2} \)
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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

−9.759633078010087114845882513777, −8.715781220519617269067992459067, −8.010216691182826770194000175682, −7.65024385460216583404467428725, −6.51519360316686909037643447562, −5.60313709408012586111736879051, −4.91471620087078538433950296498, −3.43807187385475128884232854567, −2.96215741090627392784972702705, −1.68604251927303734649373186138, 1.20230275749537431270299909491, 2.16877986192885348746052775847, 3.56203413978644367886573327063, 4.66840421544948942162143250317, 5.25775626116668737001177135895, 6.47562218671681984953798436746, 6.99329281199656667288320251889, 7.68824493082618607930019820666, 8.909757361405363848103438008072, 9.565360558033900248364380465734

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