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Properties

Label 2-1568-7.4-c1-0-39
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  + (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

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

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

\(L(1)\) \(\approx\) \(1.030963385\)
\(L(\frac12)\) \(\approx\) \(1.030963385\)
\(L(\frac{3}{2})\) 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

−8.576004917398796082028036861486, −8.101351433248015553272497434172, −7.60575065864555043943819053961, −6.72907087824218761477158480031, −5.85843125309040087153357070828, −4.61752160396890714088100843744, −3.76495749873555347166709782120, −2.38257612770313074459175624392, −1.70489761356255021185322625880, −0.33300654039931465094384668705, 2.47527336660283608049667292896, 3.30807186330713850416289422092, 3.64355245895323342070200707469, 4.84678331265803153860709497452, 5.56150485291958915231333907191, 6.85790756194090241334098235048, 7.80263662453726647587293521197, 8.332112709342950920175918656430, 9.302262233484484622109775730545, 9.806072088012232900254687635225

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