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Properties

Label 2-1575-15.8-c1-0-30
Degree $2$
Conductor $1575$
Sign $-0.749 + 0.662i$
Analytic cond. $12.5764$
Root an. cond. $3.54632$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.366 + 0.366i)2-s − 1.73i·4-s + (−0.707 + 0.707i)7-s + (1.36 − 1.36i)8-s + 2.44i·11-s + (−3.86 − 3.86i)13-s − 0.517·14-s − 2.46·16-s + (−2 − 2i)17-s + 2i·19-s + (−0.896 + 0.896i)22-s + (6.46 − 6.46i)23-s − 2.82i·26-s + (1.22 + 1.22i)28-s − 6.31·29-s + ⋯
L(s)  = 1  + (0.258 + 0.258i)2-s − 0.866i·4-s + (−0.267 + 0.267i)7-s + (0.482 − 0.482i)8-s + 0.738i·11-s + (−1.07 − 1.07i)13-s − 0.138·14-s − 0.616·16-s + (−0.485 − 0.485i)17-s + 0.458i·19-s + (−0.191 + 0.191i)22-s + (1.34 − 1.34i)23-s − 0.554i·26-s + (0.231 + 0.231i)28-s − 1.17·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.749 + 0.662i$
Analytic conductor: \(12.5764\)
Root analytic conductor: \(3.54632\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (1268, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :1/2),\ -0.749 + 0.662i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8935551754\)
\(L(\frac12)\) \(\approx\) \(0.8935551754\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 + (0.707 - 0.707i)T \)
good2 \( 1 + (-0.366 - 0.366i)T + 2iT^{2} \)
11 \( 1 - 2.44iT - 11T^{2} \)
13 \( 1 + (3.86 + 3.86i)T + 13iT^{2} \)
17 \( 1 + (2 + 2i)T + 17iT^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 + (-6.46 + 6.46i)T - 23iT^{2} \)
29 \( 1 + 6.31T + 29T^{2} \)
31 \( 1 + 4.92T + 31T^{2} \)
37 \( 1 + (0.378 - 0.378i)T - 37iT^{2} \)
41 \( 1 - 2.07iT - 41T^{2} \)
43 \( 1 + (2.82 + 2.82i)T + 43iT^{2} \)
47 \( 1 + (7.46 + 7.46i)T + 47iT^{2} \)
53 \( 1 + (-2.26 + 2.26i)T - 53iT^{2} \)
59 \( 1 - 12.6T + 59T^{2} \)
61 \( 1 + 8.92T + 61T^{2} \)
67 \( 1 + (6.31 - 6.31i)T - 67iT^{2} \)
71 \( 1 - 4.52iT - 71T^{2} \)
73 \( 1 + (-1.03 - 1.03i)T + 73iT^{2} \)
79 \( 1 + 4iT - 79T^{2} \)
83 \( 1 + (-4.53 + 4.53i)T - 83iT^{2} \)
89 \( 1 + 14.1T + 89T^{2} \)
97 \( 1 + (10.8 - 10.8i)T - 97iT^{2} \)
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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

−9.295013645409722656184736633580, −8.341376165739979440935190225106, −7.15168232164559337154109098371, −6.86862171066571922461249776574, −5.63358263652874644924128322532, −5.14632570805316390678026163316, −4.29463458095422034121784059020, −2.92869610838825802508699958635, −1.89349047819643094867716794482, −0.29629875701658051895397335808, 1.78040914435353366617880860516, 2.92306418080405153012668301245, 3.72213846859977360108511642759, 4.59048903734933999422951090321, 5.48781337097060797892861418982, 6.74984934166376284862218393505, 7.27210566421021041651061590720, 8.077334437117436605559978771330, 9.146121252216443129317303147923, 9.412394131338925404332495681821

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