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Properties

Label 2-1148-41.31-c1-0-14
Degree $2$
Conductor $1148$
Sign $0.132 + 0.991i$
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  − 2.47i·3-s + (0.848 + 2.61i)5-s + (−0.587 − 0.809i)7-s − 3.13·9-s + (0.155 + 0.0506i)11-s + (0.900 − 1.23i)13-s + (6.47 − 2.10i)15-s + (4.74 + 1.54i)17-s + (−1.52 − 2.09i)19-s + (−2.00 + 1.45i)21-s + (−3.99 − 2.89i)23-s + (−2.05 + 1.49i)25-s + 0.334i·27-s + (7.83 − 2.54i)29-s + (2.93 − 9.02i)31-s + ⋯
L(s)  = 1  − 1.43i·3-s + (0.379 + 1.16i)5-s + (−0.222 − 0.305i)7-s − 1.04·9-s + (0.0469 + 0.0152i)11-s + (0.249 − 0.343i)13-s + (1.67 − 0.542i)15-s + (1.15 + 0.374i)17-s + (−0.350 − 0.481i)19-s + (−0.437 + 0.317i)21-s + (−0.832 − 0.604i)23-s + (−0.411 + 0.299i)25-s + 0.0643i·27-s + (1.45 − 0.472i)29-s + (0.526 − 1.62i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.132 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.132 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.132 + 0.991i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ 0.132 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.694925480\)
\(L(\frac12)\) \(\approx\) \(1.694925480\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.587 + 0.809i)T \)
41 \( 1 + (-6.38 + 0.421i)T \)
good3 \( 1 + 2.47iT - 3T^{2} \)
5 \( 1 + (-0.848 - 2.61i)T + (-4.04 + 2.93i)T^{2} \)
11 \( 1 + (-0.155 - 0.0506i)T + (8.89 + 6.46i)T^{2} \)
13 \( 1 + (-0.900 + 1.23i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-4.74 - 1.54i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.52 + 2.09i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + (3.99 + 2.89i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-7.83 + 2.54i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.93 + 9.02i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.271 + 0.835i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (-2.34 - 1.70i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-0.824 + 1.13i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-0.299 + 0.0972i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-2.07 - 1.50i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (6.09 - 4.42i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-3.85 + 1.25i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (2.81 + 0.913i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + 9.41T + 73T^{2} \)
79 \( 1 - 0.527iT - 79T^{2} \)
83 \( 1 + 7.32T + 83T^{2} \)
89 \( 1 + (-4.01 - 5.52i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (-11.1 + 3.62i)T + (78.4 - 57.0i)T^{2} \)
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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

−9.859762614625177411741439656680, −8.497615155241759463790652729235, −7.74766851167824391697432618361, −7.14679346899978661132772511899, −6.18702579930678582725123122106, −6.01881466845999031080114673458, −4.26640991659630533347071013653, −2.95971331751296129426820703499, −2.23829862616690494947673958681, −0.838229578671565159200118083782, 1.33793265972238844446253497202, 3.02181170185644094969528000313, 4.00652312248563956639431475766, 4.86855797192931850024520539384, 5.43861212703292614069834144724, 6.36246841341719949640914140060, 7.78016027304123534992265586765, 8.790442621902633018352929916136, 9.103071319208084661246585939343, 10.07399038772660097872725143658

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