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Properties

Label 2-1148-7.2-c1-0-2
Degree $2$
Conductor $1148$
Sign $-0.976 + 0.213i$
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  + (1.43 + 2.48i)3-s + (−0.601 + 1.04i)5-s + (−2.46 − 0.961i)7-s + (−2.61 + 4.52i)9-s + (0.423 + 0.733i)11-s − 1.77·13-s − 3.44·15-s + (−0.665 − 1.15i)17-s + (−2.56 + 4.43i)19-s + (−1.14 − 7.49i)21-s + (0.408 − 0.706i)23-s + (1.77 + 3.07i)25-s − 6.36·27-s − 8.14·29-s + (−2.94 − 5.10i)31-s + ⋯
L(s)  = 1  + (0.827 + 1.43i)3-s + (−0.268 + 0.465i)5-s + (−0.931 − 0.363i)7-s + (−0.870 + 1.50i)9-s + (0.127 + 0.221i)11-s − 0.492·13-s − 0.889·15-s + (−0.161 − 0.279i)17-s + (−0.587 + 1.01i)19-s + (−0.250 − 1.63i)21-s + (0.0851 − 0.147i)23-s + (0.355 + 0.615i)25-s − 1.22·27-s − 1.51·29-s + (−0.528 − 0.915i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 + 0.213i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.976 + 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.976 + 0.213i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (821, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ -0.976 + 0.213i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.044196575\)
\(L(\frac12)\) \(\approx\) \(1.044196575\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.46 + 0.961i)T \)
41 \( 1 - T \)
good3 \( 1 + (-1.43 - 2.48i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.601 - 1.04i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.423 - 0.733i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.77T + 13T^{2} \)
17 \( 1 + (0.665 + 1.15i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.56 - 4.43i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.408 + 0.706i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 8.14T + 29T^{2} \)
31 \( 1 + (2.94 + 5.10i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.829 - 1.43i)T + (-18.5 - 32.0i)T^{2} \)
43 \( 1 + 3.82T + 43T^{2} \)
47 \( 1 + (2.33 - 4.04i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.82 - 3.16i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.425 - 0.737i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.56 + 7.90i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.19 - 7.26i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.66T + 71T^{2} \)
73 \( 1 + (1.54 + 2.67i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.06 - 7.03i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.90T + 83T^{2} \)
89 \( 1 + (-4.76 + 8.25i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 13.3T + 97T^{2} \)
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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

−10.05346952503390064601395082888, −9.538445947417446700995751999360, −8.883401736844779674395856524082, −7.83993491976253124242689016936, −7.07645008628696378395514508010, −5.95888000252866196744144434975, −4.84921144546042719598208463181, −3.85673607432520049735530387651, −3.42047914995331895832578388737, −2.31781270857065837516903499311, 0.38036512944821535477868626091, 1.88344617662451446120292969869, 2.80455914728205176643594834717, 3.77946099931540701206322136093, 5.18844254473756968342273614743, 6.32485500792028593766069652355, 6.93815917488840563927305907098, 7.65691624366441138836441424005, 8.733036710072074924500203158691, 8.889095835778277676285459861314

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