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Properties

Label 2-1568-7.2-c1-0-39
Degree $2$
Conductor $1568$
Sign $0.0725 - 0.997i$
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  + (−1.58 − 2.73i)3-s + (1.41 − 2.44i)5-s + (−3.5 + 6.06i)9-s + (−2.23 − 3.87i)11-s − 8.94·15-s + (2.12 + 3.67i)17-s + (−1.58 + 2.73i)19-s + (−4.47 + 7.74i)23-s + (−1.49 − 2.59i)25-s + 12.6·27-s − 6·29-s + (−3.16 − 5.47i)31-s + (−7.07 + 12.2i)33-s + (−1 + 1.73i)37-s − 1.41·41-s + ⋯
L(s)  = 1  + (−0.912 − 1.58i)3-s + (0.632 − 1.09i)5-s + (−1.16 + 2.02i)9-s + (−0.674 − 1.16i)11-s − 2.30·15-s + (0.514 + 0.891i)17-s + (−0.362 + 0.628i)19-s + (−0.932 + 1.61i)23-s + (−0.299 − 0.519i)25-s + 2.43·27-s − 1.11·29-s + (−0.567 − 0.983i)31-s + (−1.23 + 2.13i)33-s + (−0.164 + 0.284i)37-s − 0.220·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0725 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0725 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1568\)    =    \(2^{5} \cdot 7^{2}\)
Sign: $0.0725 - 0.997i$
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: \((2,\ 1568,\ (\ :1/2),\ 0.0725 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2088233948\)
\(L(\frac12)\) \(\approx\) \(0.2088233948\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.58 + 2.73i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.41 + 2.44i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.23 + 3.87i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (-2.12 - 3.67i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.58 - 2.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.47 - 7.74i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (3.16 + 5.47i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 1.41T + 41T^{2} \)
43 \( 1 + 4.47T + 43T^{2} \)
47 \( 1 + (-3.16 + 5.47i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.58 + 2.73i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.24 - 7.34i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.94T + 71T^{2} \)
73 \( 1 + (3.53 + 6.12i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.47 - 7.74i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9.48T + 83T^{2} \)
89 \( 1 + (-4.94 + 8.57i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 4.24T + 97T^{2} \)
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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

−8.490195871777270203866096947690, −8.016606943025337516236864630035, −7.27725674875207377743536979753, −5.97571027391176255269344900995, −5.82651525974922147209761393031, −5.16116935584375290178217945819, −3.61658516339395364472206719599, −2.00465511581571508080069164158, −1.37057193594114386184332767917, −0.089912856062466266356242844276, 2.32989231586122093050111391449, 3.26427510271071102421433707396, 4.39720574855794034236530334225, 5.01548524565417206157862228747, 5.85875259859724105728940705601, 6.62723901535289625122392819932, 7.42380905458112867953497069348, 8.781274368136003646510192182563, 9.672705127328570575259316993847, 10.04599086479015968962773016038

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