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Properties

Label 2-1148-41.32-c1-0-4
Degree $2$
Conductor $1148$
Sign $-0.605 - 0.795i$
Analytic cond. $9.16682$
Root an. cond. $3.02767$
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  + (0.178 − 0.178i)3-s + 0.542i·5-s + (−0.707 + 0.707i)7-s + 2.93i·9-s + (0.525 − 0.525i)11-s + (−2.31 + 2.31i)13-s + (0.0965 + 0.0965i)15-s + (0.596 + 0.596i)17-s + (−3.07 − 3.07i)19-s + 0.251i·21-s − 3.35·23-s + 4.70·25-s + (1.05 + 1.05i)27-s + (−6.56 + 6.56i)29-s − 8.61·31-s + ⋯
L(s)  = 1  + (0.102 − 0.102i)3-s + 0.242i·5-s + (−0.267 + 0.267i)7-s + 0.978i·9-s + (0.158 − 0.158i)11-s + (−0.641 + 0.641i)13-s + (0.0249 + 0.0249i)15-s + (0.144 + 0.144i)17-s + (−0.704 − 0.704i)19-s + 0.0549i·21-s − 0.700·23-s + 0.941·25-s + (0.203 + 0.203i)27-s + (−1.21 + 1.21i)29-s − 1.54·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.605 - 0.795i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9214039875\)
\(L(\frac12)\) \(\approx\) \(0.9214039875\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 0.707i)T \)
41 \( 1 + (1.28 + 6.27i)T \)
good3 \( 1 + (-0.178 + 0.178i)T - 3iT^{2} \)
5 \( 1 - 0.542iT - 5T^{2} \)
11 \( 1 + (-0.525 + 0.525i)T - 11iT^{2} \)
13 \( 1 + (2.31 - 2.31i)T - 13iT^{2} \)
17 \( 1 + (-0.596 - 0.596i)T + 17iT^{2} \)
19 \( 1 + (3.07 + 3.07i)T + 19iT^{2} \)
23 \( 1 + 3.35T + 23T^{2} \)
29 \( 1 + (6.56 - 6.56i)T - 29iT^{2} \)
31 \( 1 + 8.61T + 31T^{2} \)
37 \( 1 + 1.51T + 37T^{2} \)
43 \( 1 - 7.10iT - 43T^{2} \)
47 \( 1 + (-4.92 - 4.92i)T + 47iT^{2} \)
53 \( 1 + (3.28 - 3.28i)T - 53iT^{2} \)
59 \( 1 - 9.28T + 59T^{2} \)
61 \( 1 - 6.51iT - 61T^{2} \)
67 \( 1 + (-1.81 - 1.81i)T + 67iT^{2} \)
71 \( 1 + (6.46 - 6.46i)T - 71iT^{2} \)
73 \( 1 - 12.1iT - 73T^{2} \)
79 \( 1 + (-10.2 + 10.2i)T - 79iT^{2} \)
83 \( 1 + 7.06T + 83T^{2} \)
89 \( 1 + (2.76 - 2.76i)T - 89iT^{2} \)
97 \( 1 + (7.05 + 7.05i)T + 97iT^{2} \)
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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

−10.14122868491955436476229217905, −9.135788330490099756511715918348, −8.609913764902744672796927245811, −7.42871282338773875271037453688, −6.98017106681135029858087851178, −5.83319829201507258270602279222, −4.99833691634002301164373505481, −3.97101688580929132389976759123, −2.75067075658956440753738402278, −1.81361762274428093480179587612, 0.37323313909003264573017295699, 2.01280324460534345896068413803, 3.39716432313528319928871203936, 4.09277752372301411954316845140, 5.28217424586631049678654407534, 6.13832988180814054130476376356, 7.03722090106261790741922728191, 7.86691118299412701339979483976, 8.784946729510829815572734031564, 9.573964686232665386749031579050

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