Properties

Label 2-1148-7.4-c1-0-13
Degree $2$
Conductor $1148$
Sign $0.991 - 0.126i$
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  + (0.5 − 0.866i)3-s + (0.5 + 0.866i)5-s + (2 − 1.73i)7-s + (1 + 1.73i)9-s + (−1.5 + 2.59i)11-s + 2·13-s + 0.999·15-s + (−0.5 + 0.866i)17-s + (1.5 + 2.59i)19-s + (−0.499 − 2.59i)21-s + (2.5 + 4.33i)23-s + (2 − 3.46i)25-s + 5·27-s − 2·29-s + (−2.5 + 4.33i)31-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (0.223 + 0.387i)5-s + (0.755 − 0.654i)7-s + (0.333 + 0.577i)9-s + (−0.452 + 0.783i)11-s + 0.554·13-s + 0.258·15-s + (−0.121 + 0.210i)17-s + (0.344 + 0.596i)19-s + (−0.109 − 0.566i)21-s + (0.521 + 0.902i)23-s + (0.400 − 0.692i)25-s + 0.962·27-s − 0.371·29-s + (−0.449 + 0.777i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.991 - 0.126i$
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: \((2,\ 1148,\ (\ :1/2),\ 0.991 - 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.121520170\)
\(L(\frac12)\) \(\approx\) \(2.121520170\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 + 1.73i)T \)
41 \( 1 - T \)
good3 \( 1 + (-0.5 + 0.866i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (0.5 - 0.866i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.5 - 2.59i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.5 - 4.33i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.5 + 6.06i)T + (-18.5 + 32.0i)T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (-1.5 - 2.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.5 - 4.33i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.5 + 2.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.5 + 11.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.5 - 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + (2.5 + 4.33i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−9.961008004776927866165771651760, −8.899485584939969874396445523537, −7.937867437248048029814234220065, −7.44021761200527431332599522526, −6.74128116068101919904724702578, −5.51213348401063149214960519876, −4.66216835570596692595637984487, −3.59820411267085500824298567338, −2.25941365632885050281541916607, −1.39101956254902179472605731523, 1.06437526557409089207291463670, 2.54398730826856149381154172190, 3.57165973902465274383172110991, 4.68075855893167676331833226928, 5.39258882848160249493338012395, 6.31836680348769194072082136327, 7.39858090564515469227861968815, 8.482352520140639186940626269576, 8.899752144273314320040223909686, 9.583494187759943191617791722417

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