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Properties

Label 2-1148-287.163-c1-0-13
Degree $2$
Conductor $1148$
Sign $0.609 + 0.792i$
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  + (−2.67 + 1.54i)3-s + (−1.10 + 1.90i)5-s + (2.60 − 0.447i)7-s + (3.25 − 5.64i)9-s + (−3.16 + 1.82i)11-s − 0.353i·13-s − 6.79i·15-s + (−3.88 + 2.24i)17-s + (−5.86 − 3.38i)19-s + (−6.27 + 5.21i)21-s + (3.07 − 5.32i)23-s + (0.0757 + 0.131i)25-s + 10.8i·27-s − 0.443i·29-s + (−2.14 − 3.71i)31-s + ⋯
L(s)  = 1  + (−1.54 + 0.890i)3-s + (−0.492 + 0.852i)5-s + (0.985 − 0.169i)7-s + (1.08 − 1.88i)9-s + (−0.953 + 0.550i)11-s − 0.0979i·13-s − 1.75i·15-s + (−0.941 + 0.543i)17-s + (−1.34 − 0.776i)19-s + (−1.36 + 1.13i)21-s + (0.640 − 1.10i)23-s + (0.0151 + 0.0262i)25-s + 2.08i·27-s − 0.0823i·29-s + (−0.385 − 0.667i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.609 + 0.792i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.609 + 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.609 + 0.792i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ 0.609 + 0.792i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3450848782\)
\(L(\frac12)\) \(\approx\) \(0.3450848782\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.60 + 0.447i)T \)
41 \( 1 + (0.647 - 6.37i)T \)
good3 \( 1 + (2.67 - 1.54i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.10 - 1.90i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.16 - 1.82i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.353iT - 13T^{2} \)
17 \( 1 + (3.88 - 2.24i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.86 + 3.38i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.07 + 5.32i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.443iT - 29T^{2} \)
31 \( 1 + (2.14 + 3.71i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.730 + 1.26i)T + (-18.5 - 32.0i)T^{2} \)
43 \( 1 + 6.86T + 43T^{2} \)
47 \( 1 + (5.36 + 3.09i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-9.66 + 5.57i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.14 - 8.90i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.88 + 10.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-9.36 + 5.40i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.5iT - 71T^{2} \)
73 \( 1 + (1.23 + 2.13i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.47 + 0.852i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.57T + 83T^{2} \)
89 \( 1 + (-15.1 - 8.75i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 15.0iT - 97T^{2} \)
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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

−10.24923954885480187870653194413, −8.935801621492325456623924533006, −7.975361890740684946407768280641, −6.88508288579937187430650649919, −6.40694048831050279003194077409, −5.13924288079615393836838884006, −4.68467894541452328885561073302, −3.83884555220831252830154515636, −2.27960526184281152492303976220, −0.21886503899596912492149244565, 1.04817060808100068554722864160, 2.17703069858059675369305817167, 4.17187827147168302371711051294, 5.14517421766495467050828110118, 5.43031165101293616755625282919, 6.57331690382528388860100343411, 7.37778741937457581107320481849, 8.221526862506021490866419518061, 8.767286207812820308694634211604, 10.30677885138884545980500986516

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