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Properties

Label 2-1568-7.6-c2-0-18
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  + 3.68i·3-s − 3.04i·5-s − 4.57·9-s + 2.35·11-s + 25.3i·13-s + 11.2·15-s − 3.56i·17-s − 16.3i·19-s − 17.6·23-s + 15.7·25-s + 16.2i·27-s + 36.1·29-s − 7.22i·31-s + 8.66i·33-s + 36.8·37-s + ⋯
L(s)  = 1  + 1.22i·3-s − 0.609i·5-s − 0.508·9-s + 0.213·11-s + 1.94i·13-s + 0.748·15-s − 0.209i·17-s − 0.861i·19-s − 0.768·23-s + 0.628·25-s + 0.603i·27-s + 1.24·29-s − 0.233i·31-s + 0.262i·33-s + 0.994·37-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

\(L(\frac{3}{2})\) \(\approx\) \(1.607370571\)
\(L(\frac12)\) \(\approx\) \(1.607370571\)
\(L(2)\) 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 \( 1 - 3.68iT - 9T^{2} \)
5 \( 1 + 3.04iT - 25T^{2} \)
11 \( 1 - 2.35T + 121T^{2} \)
13 \( 1 - 25.3iT - 169T^{2} \)
17 \( 1 + 3.56iT - 289T^{2} \)
19 \( 1 + 16.3iT - 361T^{2} \)
23 \( 1 + 17.6T + 529T^{2} \)
29 \( 1 - 36.1T + 841T^{2} \)
31 \( 1 + 7.22iT - 961T^{2} \)
37 \( 1 - 36.8T + 1.36e3T^{2} \)
41 \( 1 - 53.7iT - 1.68e3T^{2} \)
43 \( 1 + 51.2T + 1.84e3T^{2} \)
47 \( 1 + 31.3iT - 2.20e3T^{2} \)
53 \( 1 + 70.2T + 2.80e3T^{2} \)
59 \( 1 - 94.0iT - 3.48e3T^{2} \)
61 \( 1 + 2.18iT - 3.72e3T^{2} \)
67 \( 1 - 24.9T + 4.48e3T^{2} \)
71 \( 1 - 50.8T + 5.04e3T^{2} \)
73 \( 1 - 79.5iT - 5.32e3T^{2} \)
79 \( 1 + 115.T + 6.24e3T^{2} \)
83 \( 1 - 154. iT - 6.88e3T^{2} \)
89 \( 1 - 114. iT - 7.92e3T^{2} \)
97 \( 1 - 53.9iT - 9.40e3T^{2} \)
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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

−9.567371494404045175435096734454, −8.944797060686528820629271593593, −8.284718598814052335044313638812, −7.00040681331663574690542851341, −6.33915914784781010042619822099, −5.05621724940112672698766827126, −4.53053179186964114379881658163, −3.94116462258678286872467920313, −2.66708505963457594617266393232, −1.29835734739492400089282097177, 0.46267100655867485747416016513, 1.58185297154597373738730246843, 2.71380380759754799556680503473, 3.51844874671694373630112898103, 4.90397122916293556134893328573, 6.02662507263964141821866652143, 6.43791061824607752084489609307, 7.46707551959848820336775949106, 7.931834824955842754989782809459, 8.611980183665736022431649593539

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