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Properties

Label 2-1148-41.32-c1-0-13
Degree $2$
Conductor $1148$
Sign $0.990 + 0.141i$
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  + (1.46 − 1.46i)3-s + 2.07i·5-s + (−0.707 + 0.707i)7-s − 1.26i·9-s + (3.13 − 3.13i)11-s + (−1.44 + 1.44i)13-s + (3.03 + 3.03i)15-s + (−2.75 − 2.75i)17-s + (3.62 + 3.62i)19-s + 2.06i·21-s + 8.09·23-s + 0.674·25-s + (2.53 + 2.53i)27-s + (6.51 − 6.51i)29-s − 6.43·31-s + ⋯
L(s)  = 1  + (0.843 − 0.843i)3-s + 0.930i·5-s + (−0.267 + 0.267i)7-s − 0.422i·9-s + (0.944 − 0.944i)11-s + (−0.400 + 0.400i)13-s + (0.784 + 0.784i)15-s + (−0.667 − 0.667i)17-s + (0.831 + 0.831i)19-s + 0.450i·21-s + 1.68·23-s + 0.134·25-s + (0.487 + 0.487i)27-s + (1.20 − 1.20i)29-s − 1.15·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.141i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.990 + 0.141i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ 0.990 + 0.141i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.266393457\)
\(L(\frac12)\) \(\approx\) \(2.266393457\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 0.707i)T \)
41 \( 1 + (-5.32 - 3.55i)T \)
good3 \( 1 + (-1.46 + 1.46i)T - 3iT^{2} \)
5 \( 1 - 2.07iT - 5T^{2} \)
11 \( 1 + (-3.13 + 3.13i)T - 11iT^{2} \)
13 \( 1 + (1.44 - 1.44i)T - 13iT^{2} \)
17 \( 1 + (2.75 + 2.75i)T + 17iT^{2} \)
19 \( 1 + (-3.62 - 3.62i)T + 19iT^{2} \)
23 \( 1 - 8.09T + 23T^{2} \)
29 \( 1 + (-6.51 + 6.51i)T - 29iT^{2} \)
31 \( 1 + 6.43T + 31T^{2} \)
37 \( 1 - 3.66T + 37T^{2} \)
43 \( 1 + 0.897iT - 43T^{2} \)
47 \( 1 + (-8.63 - 8.63i)T + 47iT^{2} \)
53 \( 1 + (7.62 - 7.62i)T - 53iT^{2} \)
59 \( 1 + 10.3T + 59T^{2} \)
61 \( 1 - 1.15iT - 61T^{2} \)
67 \( 1 + (-0.147 - 0.147i)T + 67iT^{2} \)
71 \( 1 + (-2.99 + 2.99i)T - 71iT^{2} \)
73 \( 1 + 8.52iT - 73T^{2} \)
79 \( 1 + (5.31 - 5.31i)T - 79iT^{2} \)
83 \( 1 + 13.1T + 83T^{2} \)
89 \( 1 + (-0.506 + 0.506i)T - 89iT^{2} \)
97 \( 1 + (10.8 + 10.8i)T + 97iT^{2} \)
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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

−9.403491131475447723739247530063, −9.085703444065639548297336516121, −8.025017684056034135322329816265, −7.27132287490536287216858684447, −6.66038520521670361334026250451, −5.84532521861207101463249867825, −4.42656938527782363107310300569, −3.06983181079006161278503792516, −2.71459765760350036527129946301, −1.26330505820207442683611483563, 1.14306582632646765287732598680, 2.70578520833785798982016472127, 3.71601249388370370734525713855, 4.57865110183380792943513913094, 5.16914465438293553203612359206, 6.66615236149165163936175440016, 7.34297035564386894736532465981, 8.601472943671714348036412715256, 9.069501498612904782812716728657, 9.521884908892081896176772845800

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