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Properties

Label 2-1568-7.2-c1-0-26
Degree $2$
Conductor $1568$
Sign $0.386 + 0.922i$
Analytic cond. $12.5205$
Root an. cond. $3.53843$
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  + (0.618 + 1.07i)3-s + (−1.61 + 2.80i)5-s + (0.736 − 1.27i)9-s + (−3.23 − 5.60i)11-s + 0.763·13-s − 4·15-s + (−2.23 − 3.87i)17-s + (−0.618 + 1.07i)19-s + (2 − 3.46i)23-s + (−2.73 − 4.73i)25-s + 5.52·27-s − 4.47·29-s + (−1.23 − 2.14i)31-s + (3.99 − 6.92i)33-s + (2.23 − 3.87i)37-s + ⋯
L(s)  = 1  + (0.356 + 0.618i)3-s + (−0.723 + 1.25i)5-s + (0.245 − 0.424i)9-s + (−0.975 − 1.68i)11-s + 0.211·13-s − 1.03·15-s + (−0.542 − 0.939i)17-s + (−0.141 + 0.245i)19-s + (0.417 − 0.722i)23-s + (−0.547 − 0.947i)25-s + 1.06·27-s − 0.830·29-s + (−0.222 − 0.384i)31-s + (0.696 − 1.20i)33-s + (0.367 − 0.636i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1568\)    =    \(2^{5} \cdot 7^{2}\)
Sign: $0.386 + 0.922i$
Analytic conductor: \(12.5205\)
Root analytic conductor: \(3.53843\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1568} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1568,\ (\ :1/2),\ 0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9283799455\)
\(L(\frac12)\) \(\approx\) \(0.9283799455\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.618 - 1.07i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.61 - 2.80i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.23 + 5.60i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.763T + 13T^{2} \)
17 \( 1 + (2.23 + 3.87i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.618 - 1.07i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.47T + 29T^{2} \)
31 \( 1 + (1.23 + 2.14i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.23 + 3.87i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.47T + 41T^{2} \)
43 \( 1 + 6.47T + 43T^{2} \)
47 \( 1 + (-5.23 + 9.06i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5 - 8.66i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.61 + 7.99i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.61 - 9.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.94T + 71T^{2} \)
73 \( 1 + (-1.47 - 2.54i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.47 + 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 9.23T + 83T^{2} \)
89 \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 12.4T + 97T^{2} \)
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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

−9.182001448623294826133923782041, −8.561542953801290089233846946743, −7.69435000150578282821794383907, −6.91917392386609494132689691239, −6.12472487427789030911784089386, −5.07844481449070923253486601402, −3.88617041770536397016379778328, −3.30466200013414942013579493394, −2.57289404709095337168354751736, −0.34602097072526359280066322291, 1.42797715517561468564726213099, 2.24093177878760693690444564115, 3.72614115959193451790583610042, 4.78109764087852007924899706430, 5.08219931586009969905089441816, 6.56227331910042142521843247123, 7.45788556561610143789598912212, 7.924841846604238926411104750192, 8.598345758658559653893532902415, 9.433097241075489770499640339518

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