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Properties

Label 2-1568-56.19-c1-0-35
Degree $2$
Conductor $1568$
Sign $-0.995 - 0.0956i$
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  + (0.937 − 0.541i)3-s + (0.662 − 1.14i)5-s + (−0.914 + 1.58i)9-s + (−1 − 1.73i)11-s − 5.85·13-s − 1.43i·15-s + (−3.86 + 2.23i)17-s + (−3.58 − 2.07i)19-s + (−7.24 − 4.18i)23-s + (1.62 + 2.80i)25-s + 5.22i·27-s + 4.47i·29-s + (−3.20 − 5.54i)31-s + (−1.87 − 1.08i)33-s + (−2.12 − 1.22i)37-s + ⋯
L(s)  = 1  + (0.541 − 0.312i)3-s + (0.296 − 0.513i)5-s + (−0.304 + 0.527i)9-s + (−0.301 − 0.522i)11-s − 1.62·13-s − 0.370i·15-s + (−0.936 + 0.540i)17-s + (−0.823 − 0.475i)19-s + (−1.51 − 0.871i)23-s + (0.324 + 0.561i)25-s + 1.00i·27-s + 0.831i·29-s + (−0.574 − 0.995i)31-s + (−0.326 − 0.188i)33-s + (−0.348 − 0.201i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1568\)    =    \(2^{5} \cdot 7^{2}\)
Sign: $-0.995 - 0.0956i$
Analytic conductor: \(12.5205\)
Root analytic conductor: \(3.53843\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1568} (1391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1568,\ (\ :1/2),\ -0.995 - 0.0956i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3188725648\)
\(L(\frac12)\) \(\approx\) \(0.3188725648\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.937 + 0.541i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.662 + 1.14i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.85T + 13T^{2} \)
17 \( 1 + (3.86 - 2.23i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.58 + 2.07i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (7.24 + 4.18i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.47iT - 29T^{2} \)
31 \( 1 + (3.20 + 5.54i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.12 + 1.22i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 0.317iT - 41T^{2} \)
43 \( 1 - 3.17T + 43T^{2} \)
47 \( 1 + (-6.40 + 11.0i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3 - 1.73i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.937 + 0.541i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.80 + 8.31i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6 - 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-6.12 + 3.53i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.75 + 1.01i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 5.67iT - 83T^{2} \)
89 \( 1 + (11.0 + 6.37i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.23iT - 97T^{2} \)
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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

−8.820679868934094786308263047057, −8.371515404482903138371921651918, −7.51325599649114276198957273133, −6.71846716176014519116621373427, −5.62092078041616768755501493266, −4.92071318838923920432292271701, −3.93257462366133089728975013385, −2.50626032696532254105675834434, −2.04268143573349750016100253322, −0.10008941964297490691661040854, 2.15198205152095622688720893160, 2.72285076053280675015495599217, 3.93872956910222530576610489092, 4.72434473250123299084840652024, 5.83260495581484462605002071552, 6.67107712035539428192855927493, 7.47035966725347054976250146670, 8.265412431959844252639177290788, 9.213821954850366015735864884163, 9.804650833041360397960138890644

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