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Properties

Label 2-1148-41.18-c1-0-2
Degree $2$
Conductor $1148$
Sign $-0.782 - 0.623i$
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  + 0.325·3-s + (0.538 − 0.391i)5-s + (−0.309 + 0.951i)7-s − 2.89·9-s + (−4.24 − 3.08i)11-s + (1.13 + 3.48i)13-s + (0.175 − 0.127i)15-s + (1.83 + 1.33i)17-s + (−2.09 + 6.45i)19-s + (−0.100 + 0.309i)21-s + (−2.80 − 8.64i)23-s + (−1.40 + 4.33i)25-s − 1.91·27-s + (−8.24 + 5.99i)29-s + (5.31 + 3.86i)31-s + ⋯
L(s)  = 1  + 0.187·3-s + (0.241 − 0.175i)5-s + (−0.116 + 0.359i)7-s − 0.964·9-s + (−1.28 − 0.930i)11-s + (0.313 + 0.965i)13-s + (0.0452 − 0.0329i)15-s + (0.444 + 0.322i)17-s + (−0.481 + 1.48i)19-s + (−0.0219 + 0.0675i)21-s + (−0.585 − 1.80i)23-s + (−0.281 + 0.866i)25-s − 0.369·27-s + (−1.53 + 1.11i)29-s + (0.954 + 0.693i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.782 - 0.623i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.782 - 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.782 - 0.623i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ -0.782 - 0.623i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5491399608\)
\(L(\frac12)\) \(\approx\) \(0.5491399608\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.309 - 0.951i)T \)
41 \( 1 + (-2.82 + 5.74i)T \)
good3 \( 1 - 0.325T + 3T^{2} \)
5 \( 1 + (-0.538 + 0.391i)T + (1.54 - 4.75i)T^{2} \)
11 \( 1 + (4.24 + 3.08i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-1.13 - 3.48i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.83 - 1.33i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (2.09 - 6.45i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (2.80 + 8.64i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (8.24 - 5.99i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-5.31 - 3.86i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (7.58 - 5.51i)T + (11.4 - 35.1i)T^{2} \)
43 \( 1 + (-1.39 - 4.30i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (0.991 + 3.05i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-0.495 + 0.359i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.83 - 8.71i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-3.66 + 11.2i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-8.29 + 6.02i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (2.25 + 1.64i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + 12.6T + 73T^{2} \)
79 \( 1 + 7.25T + 79T^{2} \)
83 \( 1 + 5.91T + 83T^{2} \)
89 \( 1 + (2.43 - 7.49i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (6.91 - 5.02i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.24946624393269025132363290917, −9.103273120277114783807201932946, −8.464213902356710114035270275774, −7.983486975639838993162456896909, −6.64801737233992512108041929890, −5.79734275787978915346446597878, −5.25454086787379348739005656003, −3.84864821432597657658324722684, −2.92474064551237196023194387181, −1.80656812855175581843052123797, 0.21207231702870589160268660170, 2.21962637948728595841707324957, 2.99385498874053268375294032273, 4.19249404245496286516577629438, 5.39832452424060579501327372835, 5.87281572576329303631973789696, 7.23043929458243814847267182996, 7.75799413335077563778239290492, 8.582656531557141391019722408240, 9.695062533521199917493459235213

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