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Properties

Label 2-1568-7.6-c2-0-29
Degree $2$
Conductor $1568$
Sign $0.755 + 0.654i$
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  − 5.73i·3-s + 6.30i·5-s − 23.9·9-s + 5.41·11-s − 15.9i·13-s + 36.1·15-s + 20.4i·17-s + 13.5i·19-s + 4.70·23-s − 14.7·25-s + 85.7i·27-s + 1.76·29-s + 13.7i·31-s − 31.0i·33-s + 10.4·37-s + ⋯
L(s)  = 1  − 1.91i·3-s + 1.26i·5-s − 2.66·9-s + 0.491·11-s − 1.22i·13-s + 2.41·15-s + 1.20i·17-s + 0.715i·19-s + 0.204·23-s − 0.589·25-s + 3.17i·27-s + 0.0608·29-s + 0.443i·31-s − 0.941i·33-s + 0.282·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1568\)    =    \(2^{5} \cdot 7^{2}\)
Sign: $0.755 + 0.654i$
Analytic conductor: \(42.7249\)
Root analytic conductor: \(6.53642\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1568} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1568,\ (\ :1),\ 0.755 + 0.654i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.778878109\)
\(L(\frac12)\) \(\approx\) \(1.778878109\)
\(L(2)\) 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 \( 1 + 5.73iT - 9T^{2} \)
5 \( 1 - 6.30iT - 25T^{2} \)
11 \( 1 - 5.41T + 121T^{2} \)
13 \( 1 + 15.9iT - 169T^{2} \)
17 \( 1 - 20.4iT - 289T^{2} \)
19 \( 1 - 13.5iT - 361T^{2} \)
23 \( 1 - 4.70T + 529T^{2} \)
29 \( 1 - 1.76T + 841T^{2} \)
31 \( 1 - 13.7iT - 961T^{2} \)
37 \( 1 - 10.4T + 1.36e3T^{2} \)
41 \( 1 + 11.2iT - 1.68e3T^{2} \)
43 \( 1 - 49.1T + 1.84e3T^{2} \)
47 \( 1 + 10.4iT - 2.20e3T^{2} \)
53 \( 1 - 32.1T + 2.80e3T^{2} \)
59 \( 1 - 66.1iT - 3.48e3T^{2} \)
61 \( 1 + 31.9iT - 3.72e3T^{2} \)
67 \( 1 - 98.5T + 4.48e3T^{2} \)
71 \( 1 - 61.7T + 5.04e3T^{2} \)
73 \( 1 + 18.0iT - 5.32e3T^{2} \)
79 \( 1 - 30.0T + 6.24e3T^{2} \)
83 \( 1 + 63.4iT - 6.88e3T^{2} \)
89 \( 1 - 137. iT - 7.92e3T^{2} \)
97 \( 1 + 131. iT - 9.40e3T^{2} \)
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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

−8.811944973105284360513789088843, −8.069838120214215857193166626392, −7.49551911968562249698504376212, −6.76632803559906199196899287706, −6.15645887660567215764303100532, −5.53233384137667802360865749119, −3.66246305170709548886033983610, −2.84057778215205497969069575176, −1.95398265651525923556397903206, −0.842004208848577415037848250484, 0.66950895020887725540272765594, 2.45846514935990885162674787524, 3.68410802586457233572512557592, 4.47999607249909307684126321360, 4.87442857130878490019427139375, 5.68617539013436128474368465196, 6.80919241739573745632679853869, 8.137353482091291465810492120726, 9.004555301201884429212415560871, 9.280277778588975414327598842101

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