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Properties

Label 2-1568-7.4-c1-0-22
Degree $2$
Conductor $1568$
Sign $0.749 + 0.661i$
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  + (0.707 − 1.22i)3-s + (0.500 + 0.866i)9-s + (1 − 1.73i)11-s + 2.82·13-s + (−2.12 + 3.67i)17-s + (−2.12 − 3.67i)19-s + (4 + 6.92i)23-s + (2.5 − 4.33i)25-s + 5.65·27-s + 6·29-s + (4.24 − 7.34i)31-s + (−1.41 − 2.44i)33-s + (1 + 1.73i)37-s + (2.00 − 3.46i)39-s − 4.24·41-s + ⋯
L(s)  = 1  + (0.408 − 0.707i)3-s + (0.166 + 0.288i)9-s + (0.301 − 0.522i)11-s + 0.784·13-s + (−0.514 + 0.891i)17-s + (−0.486 − 0.842i)19-s + (0.834 + 1.44i)23-s + (0.5 − 0.866i)25-s + 1.08·27-s + 1.11·29-s + (0.762 − 1.31i)31-s + (−0.246 − 0.426i)33-s + (0.164 + 0.284i)37-s + (0.320 − 0.554i)39-s − 0.662·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1568\)    =    \(2^{5} \cdot 7^{2}\)
Sign: $0.749 + 0.661i$
Analytic conductor: \(12.5205\)
Root analytic conductor: \(3.53843\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1568} (1537, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1568,\ (\ :1/2),\ 0.749 + 0.661i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.150288611\)
\(L(\frac12)\) \(\approx\) \(2.150288611\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 + 1.22i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.82T + 13T^{2} \)
17 \( 1 + (2.12 - 3.67i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.12 + 3.67i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4 - 6.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (-4.24 + 7.34i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.24T + 41T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 + (1.41 + 2.44i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.36 + 11.0i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.82 - 4.89i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6 + 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 + (-0.707 + 1.22i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6 - 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.89T + 83T^{2} \)
89 \( 1 + (2.12 + 3.67i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 18.3T + 97T^{2} \)
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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

−9.144399036367582326781533113732, −8.341230644090963448138783253506, −7.995528597450315249668584024427, −6.70727097688508719270645118009, −6.47511429185927140341434307874, −5.20969371054053976634268817770, −4.23944682552868386953655297417, −3.16976152308242916248895435519, −2.12682213912792867268792346603, −1.01695241625463350749727174725, 1.17047501144676010180972155662, 2.67665685012553851823001319563, 3.57589638830228503420089540138, 4.47343924257739265751783120442, 5.13629423247207676797421998442, 6.63635984785345843292273902586, 6.77963024594357254409385892743, 8.264231924297826497030353517519, 8.749253970132917756117112270443, 9.479678513035044839713883741057

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