Properties

Label 2-1456-13.3-c1-0-40
Degree $2$
Conductor $1456$
Sign $-0.872 + 0.488i$
Analytic cond. $11.6262$
Root an. cond. $3.40972$
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  + (−0.190 − 0.330i)3-s + 0.381·5-s + (0.5 − 0.866i)7-s + (1.42 − 2.47i)9-s + (−2.42 − 4.20i)11-s + (−2.5 − 2.59i)13-s + (−0.0729 − 0.126i)15-s + (−3.73 + 6.47i)17-s + (2.42 − 4.20i)19-s − 0.381·21-s + (2.23 + 3.87i)23-s − 4.85·25-s − 2.23·27-s + (2.04 + 3.54i)29-s − 8.70·31-s + ⋯
L(s)  = 1  + (−0.110 − 0.190i)3-s + 0.170·5-s + (0.188 − 0.327i)7-s + (0.475 − 0.823i)9-s + (−0.731 − 1.26i)11-s + (−0.693 − 0.720i)13-s + (−0.0188 − 0.0326i)15-s + (−0.906 + 1.56i)17-s + (0.556 − 0.964i)19-s − 0.0833·21-s + (0.466 + 0.807i)23-s − 0.970·25-s − 0.430·27-s + (0.379 + 0.657i)29-s − 1.56·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $-0.872 + 0.488i$
Analytic conductor: \(11.6262\)
Root analytic conductor: \(3.40972\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1456} (1121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1456,\ (\ :1/2),\ -0.872 + 0.488i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8784663408\)
\(L(\frac12)\) \(\approx\) \(0.8784663408\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (2.5 + 2.59i)T \)
good3 \( 1 + (0.190 + 0.330i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 0.381T + 5T^{2} \)
11 \( 1 + (2.42 + 4.20i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.73 - 6.47i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.42 + 4.20i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.23 - 3.87i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.04 - 3.54i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.70T + 31T^{2} \)
37 \( 1 + (2 + 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.61 + 4.53i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.78 - 6.54i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.23T + 47T^{2} \)
53 \( 1 - 8.23T + 53T^{2} \)
59 \( 1 + (-1.11 + 1.93i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3 + 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.354 - 0.613i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.09 + 7.08i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 6.70T + 83T^{2} \)
89 \( 1 + (8.04 + 13.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.07 - 10.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.146708880868851053511199202911, −8.391416448863149153516109269401, −7.52161940403805959939697673092, −6.79600763467352023239541919308, −5.82127914794909148561939263234, −5.17553296970179087253123296717, −3.90459372055797932037366676000, −3.12677367326224512025377922540, −1.73710827097335748374525934714, −0.33316712018401380513084832639, 1.91467115775109604064111656407, 2.52600830685656802271889325451, 4.14385511469237586900135048246, 4.91990889834404620433462475149, 5.42477486779351927903654911770, 6.87619493288062673956617636289, 7.31829098600882542696707811629, 8.172436796074559810854865647450, 9.253410802939160469714447755541, 9.875846761570140983939607155235

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