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Properties

Label 2-1148-41.9-c1-0-1
Degree $2$
Conductor $1148$
Sign $0.228 - 0.973i$
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  + (−1.38 − 1.38i)3-s + 2.11i·5-s + (0.707 + 0.707i)7-s + 0.852i·9-s + (−0.520 − 0.520i)11-s + (0.612 + 0.612i)13-s + (2.93 − 2.93i)15-s + (1.16 − 1.16i)17-s + (−2.68 + 2.68i)19-s − 1.96i·21-s − 3.65·23-s + 0.540·25-s + (−2.98 + 2.98i)27-s + (3.93 + 3.93i)29-s − 4.96·31-s + ⋯
L(s)  = 1  + (−0.801 − 0.801i)3-s + 0.944i·5-s + (0.267 + 0.267i)7-s + 0.284i·9-s + (−0.156 − 0.156i)11-s + (0.169 + 0.169i)13-s + (0.756 − 0.756i)15-s + (0.281 − 0.281i)17-s + (−0.616 + 0.616i)19-s − 0.428i·21-s − 0.762·23-s + 0.108·25-s + (−0.573 + 0.573i)27-s + (0.730 + 0.730i)29-s − 0.891·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.228 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.228 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.228 - 0.973i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ 0.228 - 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8575038752\)
\(L(\frac12)\) \(\approx\) \(0.8575038752\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.38 - 6.25i)T \)
good3 \( 1 + (1.38 + 1.38i)T + 3iT^{2} \)
5 \( 1 - 2.11iT - 5T^{2} \)
11 \( 1 + (0.520 + 0.520i)T + 11iT^{2} \)
13 \( 1 + (-0.612 - 0.612i)T + 13iT^{2} \)
17 \( 1 + (-1.16 + 1.16i)T - 17iT^{2} \)
19 \( 1 + (2.68 - 2.68i)T - 19iT^{2} \)
23 \( 1 + 3.65T + 23T^{2} \)
29 \( 1 + (-3.93 - 3.93i)T + 29iT^{2} \)
31 \( 1 + 4.96T + 31T^{2} \)
37 \( 1 + 1.57T + 37T^{2} \)
43 \( 1 - 8.44iT - 43T^{2} \)
47 \( 1 + (1.57 - 1.57i)T - 47iT^{2} \)
53 \( 1 + (-2.25 - 2.25i)T + 53iT^{2} \)
59 \( 1 - 6.36T + 59T^{2} \)
61 \( 1 - 8.42iT - 61T^{2} \)
67 \( 1 + (0.439 - 0.439i)T - 67iT^{2} \)
71 \( 1 + (1.13 + 1.13i)T + 71iT^{2} \)
73 \( 1 - 0.278iT - 73T^{2} \)
79 \( 1 + (3.94 + 3.94i)T + 79iT^{2} \)
83 \( 1 - 8.17T + 83T^{2} \)
89 \( 1 + (-11.2 - 11.2i)T + 89iT^{2} \)
97 \( 1 + (9.32 - 9.32i)T - 97iT^{2} \)
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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

−10.20970870429803880942912890531, −9.157460173787537606816474804470, −8.126885131060409619458864309224, −7.35431904422556831189920728397, −6.52609208233857573568683930830, −6.03631115308922574059680622086, −5.05550851432952151008188543466, −3.73605871886615701978475744411, −2.59849216362833776135793006691, −1.33649369156331754291351761632, 0.43880591624464557160992426087, 2.04668086402201015575524145590, 3.77458784523937417360888536849, 4.55293107322971715480106504299, 5.23485088977041695913625320003, 5.95957134240459167478928727451, 7.11465881617390618117042861141, 8.172302254084805064277420056856, 8.811367111257089805094754337484, 9.800158626306665676953535962841

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