Properties

Label 2-1148-41.32-c1-0-1
Degree $2$
Conductor $1148$
Sign $0.381 + 0.924i$
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  + (−2.32 + 2.32i)3-s + 4.24i·5-s + (0.707 − 0.707i)7-s − 7.82i·9-s + (−4.19 + 4.19i)11-s + (3.56 − 3.56i)13-s + (−9.88 − 9.88i)15-s + (−1.77 − 1.77i)17-s + (−2.53 − 2.53i)19-s + 3.29i·21-s − 2.95·23-s − 13.0·25-s + (11.2 + 11.2i)27-s + (−1.24 + 1.24i)29-s − 0.673·31-s + ⋯
L(s)  = 1  + (−1.34 + 1.34i)3-s + 1.89i·5-s + (0.267 − 0.267i)7-s − 2.60i·9-s + (−1.26 + 1.26i)11-s + (0.989 − 0.989i)13-s + (−2.55 − 2.55i)15-s + (−0.430 − 0.430i)17-s + (−0.580 − 0.580i)19-s + 0.718i·21-s − 0.615·23-s − 2.60·25-s + (2.16 + 2.16i)27-s + (−0.230 + 0.230i)29-s − 0.121·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.381 + 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.381 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.381 + 0.924i$
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: \((2,\ 1148,\ (\ :1/2),\ 0.381 + 0.924i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.07663226551\)
\(L(\frac12)\) \(\approx\) \(0.07663226551\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.364 + 6.39i)T \)
good3 \( 1 + (2.32 - 2.32i)T - 3iT^{2} \)
5 \( 1 - 4.24iT - 5T^{2} \)
11 \( 1 + (4.19 - 4.19i)T - 11iT^{2} \)
13 \( 1 + (-3.56 + 3.56i)T - 13iT^{2} \)
17 \( 1 + (1.77 + 1.77i)T + 17iT^{2} \)
19 \( 1 + (2.53 + 2.53i)T + 19iT^{2} \)
23 \( 1 + 2.95T + 23T^{2} \)
29 \( 1 + (1.24 - 1.24i)T - 29iT^{2} \)
31 \( 1 + 0.673T + 31T^{2} \)
37 \( 1 - 0.962T + 37T^{2} \)
43 \( 1 + 2.97iT - 43T^{2} \)
47 \( 1 + (-5.82 - 5.82i)T + 47iT^{2} \)
53 \( 1 + (3.22 - 3.22i)T - 53iT^{2} \)
59 \( 1 - 1.27T + 59T^{2} \)
61 \( 1 - 0.789iT - 61T^{2} \)
67 \( 1 + (0.698 + 0.698i)T + 67iT^{2} \)
71 \( 1 + (4.91 - 4.91i)T - 71iT^{2} \)
73 \( 1 + 9.13iT - 73T^{2} \)
79 \( 1 + (-7.06 + 7.06i)T - 79iT^{2} \)
83 \( 1 + 13.7T + 83T^{2} \)
89 \( 1 + (5.05 - 5.05i)T - 89iT^{2} \)
97 \( 1 + (-5.75 - 5.75i)T + 97iT^{2} \)
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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

−10.49544727673800627181782710454, −10.21192397448201160345313832325, −9.209005957008562487244951822427, −7.74601540850140720045416553920, −6.97995218112870068265481834479, −6.15860001482376421659957467118, −5.42948547442436285456531498458, −4.45068318818896386819025980188, −3.58916459454357835381760827834, −2.53044922865836345599626656751, 0.04305225069364963341703280947, 1.19749410772297993954019857317, 2.04096635336505941500134411338, 4.20668563815486199474650320796, 5.11388072752809964347712825265, 5.84085705662304576573926786101, 6.22585777014554342592999419244, 7.61261992699283225230590803108, 8.404183316454272123877895393362, 8.620735622423105275059460532937

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