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Properties

Label 2-1148-287.163-c1-0-17
Degree $2$
Conductor $1148$
Sign $0.984 + 0.174i$
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  + (1.90 − 1.09i)3-s + (−1.32 + 2.30i)5-s + (2.55 − 0.693i)7-s + (0.913 − 1.58i)9-s + (4.86 − 2.81i)11-s − 0.607i·13-s + 5.83i·15-s + (−1.42 + 0.820i)17-s + (1.82 + 1.05i)19-s + (4.09 − 4.12i)21-s + (−3.27 + 5.66i)23-s + (−1.02 − 1.78i)25-s + 2.57i·27-s − 8.24i·29-s + (−0.755 − 1.30i)31-s + ⋯
L(s)  = 1  + (1.09 − 0.634i)3-s + (−0.594 + 1.02i)5-s + (0.965 − 0.261i)7-s + (0.304 − 0.527i)9-s + (1.46 − 0.847i)11-s − 0.168i·13-s + 1.50i·15-s + (−0.344 + 0.199i)17-s + (0.419 + 0.242i)19-s + (0.894 − 0.899i)21-s + (−0.681 + 1.18i)23-s + (−0.205 − 0.356i)25-s + 0.495i·27-s − 1.53i·29-s + (−0.135 − 0.234i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.174i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.174i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.984 + 0.174i$
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: \((2,\ 1148,\ (\ :1/2),\ 0.984 + 0.174i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.541379817\)
\(L(\frac12)\) \(\approx\) \(2.541379817\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.55 + 0.693i)T \)
41 \( 1 + (-3.30 - 5.48i)T \)
good3 \( 1 + (-1.90 + 1.09i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.32 - 2.30i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.86 + 2.81i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.607iT - 13T^{2} \)
17 \( 1 + (1.42 - 0.820i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.82 - 1.05i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.27 - 5.66i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 8.24iT - 29T^{2} \)
31 \( 1 + (0.755 + 1.30i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.09 + 1.89i)T + (-18.5 - 32.0i)T^{2} \)
43 \( 1 - 9.42T + 43T^{2} \)
47 \( 1 + (3.70 + 2.13i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.13 + 2.38i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.353 + 0.612i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.16 - 5.47i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.79 + 5.07i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.07iT - 71T^{2} \)
73 \( 1 + (5.47 + 9.47i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.65 + 2.68i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.48T + 83T^{2} \)
89 \( 1 + (4.05 + 2.34i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.0178iT - 97T^{2} \)
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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

−9.581889520415540170285343786273, −8.801216949654113813621773757420, −7.88198145413881954031791652106, −7.62471624548529632463460714832, −6.67115254949394179737565611193, −5.74504356459027399872193605063, −4.12178583489608634188999111206, −3.54451551775443702105096057622, −2.46440187284802832425146604941, −1.32560486603264584445021253882, 1.30057567321926379714928993870, 2.52382550659098206312985661497, 3.98736743197645708267203193067, 4.33083066501758348747442204990, 5.19532104049382919961940277706, 6.63557408464074288429171595087, 7.63977559115619274128306246972, 8.465001702598042100270869127845, 8.980538346754049676849114714121, 9.400590602759456070106502934834

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