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Properties

Label 2-1148-7.4-c1-0-17
Degree $2$
Conductor $1148$
Sign $0.853 + 0.521i$
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  + (1.05 − 1.82i)3-s + (1.46 + 2.54i)5-s + (2.64 − 0.102i)7-s + (−0.711 − 1.23i)9-s + (2.90 − 5.02i)11-s − 1.53·13-s + 6.17·15-s + (−2.84 + 4.92i)17-s + (−1.65 − 2.86i)19-s + (2.59 − 4.92i)21-s + (1.45 + 2.51i)23-s + (−1.80 + 3.13i)25-s + 3.31·27-s − 1.53·29-s + (3.33 − 5.76i)31-s + ⋯
L(s)  = 1  + (0.607 − 1.05i)3-s + (0.656 + 1.13i)5-s + (0.999 − 0.0387i)7-s + (−0.237 − 0.410i)9-s + (0.875 − 1.51i)11-s − 0.425·13-s + 1.59·15-s + (−0.690 + 1.19i)17-s + (−0.379 − 0.656i)19-s + (0.565 − 1.07i)21-s + (0.303 + 0.525i)23-s + (−0.361 + 0.626i)25-s + 0.638·27-s − 0.285·29-s + (0.598 − 1.03i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.853 + 0.521i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.853 + 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.853 + 0.521i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ 0.853 + 0.521i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.587661190\)
\(L(\frac12)\) \(\approx\) \(2.587661190\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.64 + 0.102i)T \)
41 \( 1 - T \)
good3 \( 1 + (-1.05 + 1.82i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.46 - 2.54i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.90 + 5.02i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.53T + 13T^{2} \)
17 \( 1 + (2.84 - 4.92i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.65 + 2.86i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.45 - 2.51i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.53T + 29T^{2} \)
31 \( 1 + (-3.33 + 5.76i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.67 - 9.82i)T + (-18.5 + 32.0i)T^{2} \)
43 \( 1 + 2.99T + 43T^{2} \)
47 \( 1 + (4.48 + 7.77i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.43 - 2.49i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.87 - 8.45i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.28 + 5.68i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.17 - 7.23i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.2T + 71T^{2} \)
73 \( 1 + (1.01 - 1.75i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.15 + 8.93i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.68T + 83T^{2} \)
89 \( 1 + (-2.40 - 4.16i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 4.61T + 97T^{2} \)
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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

−9.683501032713337250286858188993, −8.570799829501255677800017870716, −8.202369412272465672330346543534, −7.17638596484593453397490840518, −6.49018902849467586631104093437, −5.86395746512718004887272684376, −4.43312971633502187290956534818, −3.15269246528844365409759310223, −2.28244193828532667806018655921, −1.33798107659015906016964354010, 1.43716854723840830600126745371, 2.46758787604752786415621510646, 4.07077212644061623437511264621, 4.70813742718271093813622727470, 5.10258950308942680569202890327, 6.53202728868329633238089069734, 7.57099691123751233371773993481, 8.553231949600302978625192373229, 9.306031008012579772834478659254, 9.505056391563856315016470791911

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