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Properties

Label 2-1148-287.163-c1-0-20
Degree $2$
Conductor $1148$
Sign $0.756 + 0.653i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 + 0.653i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.756 + 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.756 + 0.653i$
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: \((2,\ 1148,\ (\ :1/2),\ 0.756 + 0.653i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.552479821\)
\(L(\frac12)\) \(\approx\) \(2.552479821\)
\(L(\frac{3}{2})\) 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

−9.445250987183259473557490744898, −8.915415629477619599628598287087, −7.972730057763592491194043349331, −7.22038138670272033436613202883, −6.77685422287624421290198643830, −5.76952927245149443718370513279, −3.85972341216375036309954273800, −3.27272065457614774591683240410, −2.66152109644101503237157283409, −1.13274911511317467033660126038, 1.40684293306460222613935732549, 3.07721728021765791427657224318, 3.55116963500374521964543891204, 4.45721002128719565156236838249, 5.32058409271506002749077251819, 6.84239186724146117011208273204, 7.64766948224464118834692327615, 8.441762122357626640986801790952, 9.239373196349482931989755267338, 9.590580615318022192744515765071

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