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Properties

Label 2-1148-7.2-c1-0-20
Degree $2$
Conductor $1148$
Sign $-0.138 + 0.990i$
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  + (−0.224 − 0.389i)3-s + (0.393 − 0.682i)5-s + (1.75 − 1.97i)7-s + (1.39 − 2.42i)9-s + (−1.55 − 2.68i)11-s + 5.59·13-s − 0.354·15-s + (−2.69 − 4.66i)17-s + (−2.41 + 4.17i)19-s + (−1.16 − 0.239i)21-s + (−3.51 + 6.08i)23-s + (2.18 + 3.79i)25-s − 2.60·27-s − 7.52·29-s + (−2.45 − 4.24i)31-s + ⋯
L(s)  = 1  + (−0.129 − 0.224i)3-s + (0.176 − 0.305i)5-s + (0.664 − 0.747i)7-s + (0.466 − 0.807i)9-s + (−0.467 − 0.810i)11-s + 1.55·13-s − 0.0914·15-s + (−0.652 − 1.13i)17-s + (−0.553 + 0.958i)19-s + (−0.254 − 0.0522i)21-s + (−0.732 + 1.26i)23-s + (0.437 + 0.758i)25-s − 0.501·27-s − 1.39·29-s + (−0.440 − 0.763i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.138 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.138 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.138 + 0.990i$
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: \((2,\ 1148,\ (\ :1/2),\ -0.138 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.644486274\)
\(L(\frac12)\) \(\approx\) \(1.644486274\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.75 + 1.97i)T \)
41 \( 1 + T \)
good3 \( 1 + (0.224 + 0.389i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.393 + 0.682i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.55 + 2.68i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.59T + 13T^{2} \)
17 \( 1 + (2.69 + 4.66i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.41 - 4.17i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.51 - 6.08i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.52T + 29T^{2} \)
31 \( 1 + (2.45 + 4.24i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.60 + 7.97i)T + (-18.5 - 32.0i)T^{2} \)
43 \( 1 - 11.7T + 43T^{2} \)
47 \( 1 + (-2.78 + 4.81i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.97 + 5.15i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.10 - 8.83i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.44 - 4.23i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.87 + 10.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.39T + 71T^{2} \)
73 \( 1 + (-2.14 - 3.70i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.05 - 7.01i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 2.10T + 83T^{2} \)
89 \( 1 + (-8.08 + 13.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 2.23T + 97T^{2} \)
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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

−9.405750314471835836590731085453, −8.875350747391559690701407003045, −7.79196276936542200921376739447, −7.25917202233165727809076265261, −6.03523304977789831877287941588, −5.54860221225540594321156101966, −4.11757717061208309488976082989, −3.60300294862028266975482084027, −1.84280532755516955778721226314, −0.75554905123368998874587127497, 1.74411558945699350322651877653, 2.54816766483768336444885383327, 4.16105300577355230361503270181, 4.73907648255859315275996494221, 5.86954553277538372839839813096, 6.53643550008872531585563115587, 7.69362380746608373401847061375, 8.430066751363238937128457699151, 9.073281739210463497724047773460, 10.25649105054044022349578089708

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