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Properties

Label 2-1148-41.31-c1-0-11
Degree $2$
Conductor $1148$
Sign $0.859 + 0.511i$
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  + 2.90i·3-s + (−0.726 − 2.23i)5-s + (−0.587 − 0.809i)7-s − 5.44·9-s + (−4.77 − 1.55i)11-s + (0.248 − 0.341i)13-s + (6.49 − 2.11i)15-s + (5.89 + 1.91i)17-s + (0.733 + 1.00i)19-s + (2.35 − 1.70i)21-s + (−2.51 − 1.82i)23-s + (−0.426 + 0.310i)25-s − 7.10i·27-s + (7.39 − 2.40i)29-s + (1.65 − 5.09i)31-s + ⋯
L(s)  = 1  + 1.67i·3-s + (−0.324 − 0.999i)5-s + (−0.222 − 0.305i)7-s − 1.81·9-s + (−1.43 − 0.467i)11-s + (0.0688 − 0.0947i)13-s + (1.67 − 0.545i)15-s + (1.43 + 0.464i)17-s + (0.168 + 0.231i)19-s + (0.513 − 0.372i)21-s + (−0.524 − 0.381i)23-s + (−0.0853 + 0.0620i)25-s − 1.36i·27-s + (1.37 − 0.446i)29-s + (0.297 − 0.914i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.859 + 0.511i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ 0.859 + 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.039949292\)
\(L(\frac12)\) \(\approx\) \(1.039949292\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.587 + 0.809i)T \)
41 \( 1 + (-3.63 + 5.27i)T \)
good3 \( 1 - 2.90iT - 3T^{2} \)
5 \( 1 + (0.726 + 2.23i)T + (-4.04 + 2.93i)T^{2} \)
11 \( 1 + (4.77 + 1.55i)T + (8.89 + 6.46i)T^{2} \)
13 \( 1 + (-0.248 + 0.341i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-5.89 - 1.91i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-0.733 - 1.00i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + (2.51 + 1.82i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-7.39 + 2.40i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.65 + 5.09i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (3.61 + 11.1i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (-3.80 - 2.76i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-4.46 + 6.15i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-8.91 + 2.89i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (9.13 + 6.63i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (4.00 - 2.90i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (8.17 - 2.65i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (3.69 + 1.19i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 - 0.525T + 73T^{2} \)
79 \( 1 - 12.5iT - 79T^{2} \)
83 \( 1 - 1.32T + 83T^{2} \)
89 \( 1 + (5.80 + 7.99i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (7.11 - 2.31i)T + (78.4 - 57.0i)T^{2} \)
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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

−9.928793428492064040299109657455, −8.952456945899616267402829110781, −8.294377678892281036321486438024, −7.60624334930919674780635285720, −5.82946406541550522249747106864, −5.39305591805433121926466541210, −4.43621235399667457572915776507, −3.78881187110488836334230531031, −2.73887967005814927513560835128, −0.49302883089683876155620739460, 1.26537077620965533489186705883, 2.73336371359947530574020622421, 3.02041985076521006801438661840, 4.94434032361888038016080089889, 5.93326661623013789117693500228, 6.69524742249804029992840440634, 7.53208330209543811995573335548, 7.76494136426820466837401209404, 8.785313830039338853574954389547, 10.09064484167295066230172129630

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