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Properties

Label 2-1148-41.10-c1-0-16
Degree $2$
Conductor $1148$
Sign $-0.998 + 0.0476i$
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  − 1.83·3-s + (−0.827 − 2.54i)5-s + (0.809 − 0.587i)7-s + 0.370·9-s + (1.94 − 6.00i)11-s + (0.351 + 0.255i)13-s + (1.51 + 4.67i)15-s + (0.180 − 0.554i)17-s + (−2.50 + 1.81i)19-s + (−1.48 + 1.07i)21-s + (1.79 + 1.30i)23-s + (−1.76 + 1.27i)25-s + 4.82·27-s + (−3.02 − 9.30i)29-s + (−0.0253 + 0.0781i)31-s + ⋯
L(s)  = 1  − 1.05·3-s + (−0.370 − 1.13i)5-s + (0.305 − 0.222i)7-s + 0.123·9-s + (0.587 − 1.80i)11-s + (0.0975 + 0.0708i)13-s + (0.392 + 1.20i)15-s + (0.0436 − 0.134i)17-s + (−0.573 + 0.416i)19-s + (−0.324 + 0.235i)21-s + (0.375 + 0.272i)23-s + (−0.352 + 0.255i)25-s + 0.929·27-s + (−0.561 − 1.72i)29-s + (−0.00455 + 0.0140i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0476i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0476i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.998 + 0.0476i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (953, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ -0.998 + 0.0476i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5650254857\)
\(L(\frac12)\) \(\approx\) \(0.5650254857\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.809 + 0.587i)T \)
41 \( 1 + (1.88 - 6.12i)T \)
good3 \( 1 + 1.83T + 3T^{2} \)
5 \( 1 + (0.827 + 2.54i)T + (-4.04 + 2.93i)T^{2} \)
11 \( 1 + (-1.94 + 6.00i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-0.351 - 0.255i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.180 + 0.554i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (2.50 - 1.81i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-1.79 - 1.30i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (3.02 + 9.30i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.0253 - 0.0781i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-3.69 - 11.3i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (8.95 + 6.50i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (6.20 + 4.50i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.110 - 0.340i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (0.818 + 0.594i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (9.84 - 7.15i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (4.33 + 13.3i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-0.898 + 2.76i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + 0.150T + 73T^{2} \)
79 \( 1 - 3.22T + 79T^{2} \)
83 \( 1 + 5.55T + 83T^{2} \)
89 \( 1 + (12.6 - 9.18i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-3.68 - 11.3i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.320467828681570436788354378290, −8.393701235571788306270813980732, −8.093229283903695534623081285438, −6.56883837230103175082722091611, −5.98376818939775051830575494207, −5.13778126659087234685677835724, −4.36161691735445860130687701598, −3.27428687530097601392936521574, −1.31200569349370178781844062170, −0.30811365925545578896645498968, 1.76685006509697937703814346944, 3.04089250633618855668824939747, 4.29387780790127058271551292845, 5.08167967676204939303300062185, 6.11305129443201316863062778801, 6.97158951063818579151081278055, 7.29017662326255188799926273248, 8.598657519520196481847417928658, 9.541919624896606358004553168957, 10.46834344665242845830026455655

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