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Properties

Label 2-1148-41.10-c1-0-11
Degree $2$
Conductor $1148$
Sign $-0.315 + 0.948i$
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  − 1.67·3-s + (0.746 + 2.29i)5-s + (−0.809 + 0.587i)7-s − 0.210·9-s + (1.38 − 4.27i)11-s + (−1.82 − 1.32i)13-s + (−1.24 − 3.83i)15-s + (−2.33 + 7.18i)17-s + (−2.63 + 1.91i)19-s + (1.35 − 0.981i)21-s + (−3.28 − 2.38i)23-s + (−0.670 + 0.486i)25-s + 5.36·27-s + (−0.849 − 2.61i)29-s + (2.69 − 8.29i)31-s + ⋯
L(s)  = 1  − 0.964·3-s + (0.333 + 1.02i)5-s + (−0.305 + 0.222i)7-s − 0.0701·9-s + (0.418 − 1.28i)11-s + (−0.506 − 0.367i)13-s + (−0.321 − 0.990i)15-s + (−0.566 + 1.74i)17-s + (−0.605 + 0.439i)19-s + (0.294 − 0.214i)21-s + (−0.685 − 0.498i)23-s + (−0.134 + 0.0973i)25-s + 1.03·27-s + (−0.157 − 0.485i)29-s + (0.483 − 1.48i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.315 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.315 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.315 + 0.948i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (953, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ -0.315 + 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3741225426\)
\(L(\frac12)\) \(\approx\) \(0.3741225426\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.809 - 0.587i)T \)
41 \( 1 + (5.03 + 3.96i)T \)
good3 \( 1 + 1.67T + 3T^{2} \)
5 \( 1 + (-0.746 - 2.29i)T + (-4.04 + 2.93i)T^{2} \)
11 \( 1 + (-1.38 + 4.27i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (1.82 + 1.32i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (2.33 - 7.18i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (2.63 - 1.91i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (3.28 + 2.38i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (0.849 + 2.61i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-2.69 + 8.29i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.260 + 0.802i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (-3.64 - 2.64i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-4.94 - 3.59i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (3.62 + 11.1i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (8.67 + 6.30i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-5.11 + 3.71i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (0.621 + 1.91i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-3.55 + 10.9i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + 13.4T + 73T^{2} \)
79 \( 1 + 11.8T + 79T^{2} \)
83 \( 1 - 1.72T + 83T^{2} \)
89 \( 1 + (-4.18 + 3.04i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (2.47 + 7.61i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.819316297297082055698143965599, −8.646742718022839763208978620836, −7.980498034591777923857259337785, −6.60363771218603199231495871607, −6.14371509878628481220829844399, −5.74422383166042721430249560441, −4.29909684626075289159972046584, −3.25823192314921846460125891322, −2.14267926132513298600084474488, −0.19072161721090562544673347328, 1.29176710308876266024530722089, 2.67925333545914113920040347442, 4.42706993473569665966768520717, 4.85292083443411336593988641730, 5.68088247503449144128926810536, 6.81476297141572050809190523224, 7.22254444654209889845532955657, 8.686580055304848310611958220722, 9.257231602893812383325975335928, 9.987261443079934454416812445835

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