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Properties

Label 2-1568-8.3-c2-0-71
Degree $2$
Conductor $1568$
Sign $0.394 + 0.918i$
Analytic cond. $42.7249$
Root an. cond. $6.53642$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.33·3-s − 2.15i·5-s + 19.4·9-s − 5.25·11-s − 21.4i·13-s − 11.5i·15-s − 0.926·17-s − 5.93·19-s − 8.68i·23-s + 20.3·25-s + 55.6·27-s − 9.42i·29-s − 34.5i·31-s − 28.0·33-s − 12.8i·37-s + ⋯
L(s)  = 1  + 1.77·3-s − 0.431i·5-s + 2.15·9-s − 0.478·11-s − 1.64i·13-s − 0.766i·15-s − 0.0545·17-s − 0.312·19-s − 0.377i·23-s + 0.813·25-s + 2.06·27-s − 0.324i·29-s − 1.11i·31-s − 0.849·33-s − 0.345i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.394 + 0.918i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.394 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1568\)    =    \(2^{5} \cdot 7^{2}\)
Sign: $0.394 + 0.918i$
Analytic conductor: \(42.7249\)
Root analytic conductor: \(6.53642\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1568} (687, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1568,\ (\ :1),\ 0.394 + 0.918i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.750036841\)
\(L(\frac12)\) \(\approx\) \(3.750036841\)
\(L(2)\) 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 \( 1 - 5.33T + 9T^{2} \)
5 \( 1 + 2.15iT - 25T^{2} \)
11 \( 1 + 5.25T + 121T^{2} \)
13 \( 1 + 21.4iT - 169T^{2} \)
17 \( 1 + 0.926T + 289T^{2} \)
19 \( 1 + 5.93T + 361T^{2} \)
23 \( 1 + 8.68iT - 529T^{2} \)
29 \( 1 + 9.42iT - 841T^{2} \)
31 \( 1 + 34.5iT - 961T^{2} \)
37 \( 1 + 12.8iT - 1.36e3T^{2} \)
41 \( 1 + 43.1T + 1.68e3T^{2} \)
43 \( 1 - 41.7T + 1.84e3T^{2} \)
47 \( 1 - 45.9iT - 2.20e3T^{2} \)
53 \( 1 + 74.4iT - 2.80e3T^{2} \)
59 \( 1 - 53.6T + 3.48e3T^{2} \)
61 \( 1 - 27.8iT - 3.72e3T^{2} \)
67 \( 1 - 78.4T + 4.48e3T^{2} \)
71 \( 1 - 74.5iT - 5.04e3T^{2} \)
73 \( 1 + 33.6T + 5.32e3T^{2} \)
79 \( 1 - 30.2iT - 6.24e3T^{2} \)
83 \( 1 + 72.9T + 6.88e3T^{2} \)
89 \( 1 - 54.8T + 7.92e3T^{2} \)
97 \( 1 - 53.7T + 9.40e3T^{2} \)
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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

−8.875445543481910837735012561975, −8.320509168146657432281574882779, −7.82030792295794253380310934624, −7.01321004514506408641621171642, −5.74857509270626492033263557247, −4.74883737282121118537664785664, −3.78845760708855034624380674393, −2.91755405259287685301112606537, −2.20570783616331071474962106851, −0.791794856812645715702525916067, 1.56602600211197509673817923541, 2.41698511169193025268244464073, 3.27165030779780969772924121247, 4.08765375232782485773894992645, 5.00496258864632364513597518976, 6.55897127058557850458182838392, 7.09314110196542679519617488521, 7.88044779548810716639555876969, 8.807498680844157599110938174468, 9.065448562759853390722777016549

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