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Properties

Label 2-1148-287.163-c1-0-7
Degree $2$
Conductor $1148$
Sign $0.193 - 0.981i$
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  + (0.388 − 0.224i)3-s + (−0.254 + 0.440i)5-s + (−2.49 − 0.888i)7-s + (−1.39 + 2.42i)9-s + (0.882 − 0.509i)11-s − 1.40i·13-s + 0.228i·15-s + (1.94 − 1.12i)17-s + (4.34 + 2.50i)19-s + (−1.16 + 0.213i)21-s + (−1.64 + 2.84i)23-s + (2.37 + 4.10i)25-s + 2.60i·27-s + 6.91i·29-s + (1.54 + 2.67i)31-s + ⋯
L(s)  = 1  + (0.224 − 0.129i)3-s + (−0.113 + 0.197i)5-s + (−0.941 − 0.335i)7-s + (−0.466 + 0.807i)9-s + (0.266 − 0.153i)11-s − 0.390i·13-s + 0.0589i·15-s + (0.470 − 0.271i)17-s + (0.995 + 0.574i)19-s + (−0.254 + 0.0466i)21-s + (−0.343 + 0.594i)23-s + (0.474 + 0.821i)25-s + 0.500i·27-s + 1.28i·29-s + (0.277 + 0.481i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.193 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.193 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.193 - 0.981i$
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: \((2,\ 1148,\ (\ :1/2),\ 0.193 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.203809125\)
\(L(\frac12)\) \(\approx\) \(1.203809125\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.49 + 0.888i)T \)
41 \( 1 + (2.71 - 5.79i)T \)
good3 \( 1 + (-0.388 + 0.224i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.254 - 0.440i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.882 + 0.509i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.40iT - 13T^{2} \)
17 \( 1 + (-1.94 + 1.12i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.34 - 2.50i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.64 - 2.84i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.91iT - 29T^{2} \)
31 \( 1 + (-1.54 - 2.67i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.57 - 6.19i)T + (-18.5 - 32.0i)T^{2} \)
43 \( 1 + 4.41T + 43T^{2} \)
47 \( 1 + (2.96 + 1.71i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.47 + 3.16i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.98 - 5.17i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.80 - 10.0i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.50 + 3.75i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.32iT - 71T^{2} \)
73 \( 1 + (2.33 + 4.04i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (10.0 + 5.77i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.00T + 83T^{2} \)
89 \( 1 + (-5.78 - 3.33i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.65iT - 97T^{2} \)
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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

−10.04913987955756967616618752653, −9.180590457605554087506287773739, −8.300857792550559080766077982143, −7.46846681675975558460192302698, −6.80417496844412810940025806761, −5.71084711473715639052273742772, −4.96401545414219883445624823410, −3.42462170739792335828874993469, −3.05976848353926285304074306821, −1.39889586139605814606663730786, 0.53392339170131955453272729203, 2.38006793205250629091575591216, 3.38637949056105551950804479564, 4.21152402220923077409685005704, 5.47089636787337205398109618244, 6.30775800077730747703469980282, 6.98885697283434705171316977419, 8.156517500165202110393900129591, 8.906757187810327614522566757318, 9.584693513492512123818082991715

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