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Properties

Label 2-1148-41.9-c1-0-7
Degree $2$
Conductor $1148$
Sign $0.970 - 0.241i$
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  + (0.575 + 0.575i)3-s + 0.551i·5-s + (−0.707 − 0.707i)7-s − 2.33i·9-s + (−0.935 − 0.935i)11-s + (5.08 + 5.08i)13-s + (−0.317 + 0.317i)15-s + (2.38 − 2.38i)17-s + (−1.12 + 1.12i)19-s − 0.814i·21-s − 2.53·23-s + 4.69·25-s + (3.07 − 3.07i)27-s + (2.24 + 2.24i)29-s + 5.96·31-s + ⋯
L(s)  = 1  + (0.332 + 0.332i)3-s + 0.246i·5-s + (−0.267 − 0.267i)7-s − 0.779i·9-s + (−0.282 − 0.282i)11-s + (1.41 + 1.41i)13-s + (−0.0819 + 0.0819i)15-s + (0.578 − 0.578i)17-s + (−0.258 + 0.258i)19-s − 0.177i·21-s − 0.527·23-s + 0.939·25-s + (0.591 − 0.591i)27-s + (0.417 + 0.417i)29-s + 1.07·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.970 - 0.241i$
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: \((2,\ 1148,\ (\ :1/2),\ 0.970 - 0.241i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.894028771\)
\(L(\frac12)\) \(\approx\) \(1.894028771\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.93 + 4.08i)T \)
good3 \( 1 + (-0.575 - 0.575i)T + 3iT^{2} \)
5 \( 1 - 0.551iT - 5T^{2} \)
11 \( 1 + (0.935 + 0.935i)T + 11iT^{2} \)
13 \( 1 + (-5.08 - 5.08i)T + 13iT^{2} \)
17 \( 1 + (-2.38 + 2.38i)T - 17iT^{2} \)
19 \( 1 + (1.12 - 1.12i)T - 19iT^{2} \)
23 \( 1 + 2.53T + 23T^{2} \)
29 \( 1 + (-2.24 - 2.24i)T + 29iT^{2} \)
31 \( 1 - 5.96T + 31T^{2} \)
37 \( 1 + 2.58T + 37T^{2} \)
43 \( 1 - 2.93iT - 43T^{2} \)
47 \( 1 + (-7.00 + 7.00i)T - 47iT^{2} \)
53 \( 1 + (-1.30 - 1.30i)T + 53iT^{2} \)
59 \( 1 - 5.56T + 59T^{2} \)
61 \( 1 - 10.0iT - 61T^{2} \)
67 \( 1 + (2.98 - 2.98i)T - 67iT^{2} \)
71 \( 1 + (-4.70 - 4.70i)T + 71iT^{2} \)
73 \( 1 - 4.34iT - 73T^{2} \)
79 \( 1 + (3.27 + 3.27i)T + 79iT^{2} \)
83 \( 1 + 14.5T + 83T^{2} \)
89 \( 1 + (4.61 + 4.61i)T + 89iT^{2} \)
97 \( 1 + (-8.49 + 8.49i)T - 97iT^{2} \)
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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

−9.864971944670510941957721644535, −8.862547938132934911425822422000, −8.544078418710489584952419670371, −7.20933561605400672171498822004, −6.53455946900411555710132933654, −5.74015043704210317391296642332, −4.35361470057907617939764966681, −3.69584142405019886535253657846, −2.72072908781914356613369417858, −1.10020674097546459621684406876, 1.08769422358885116110153484544, 2.46710359403723522617753180057, 3.38674027594031057590141445302, 4.62265317811803927590122542218, 5.61263282774327545676850211099, 6.32142098496889658041358266919, 7.50133749953218078801554569165, 8.259352577673131240481013998382, 8.618695140498819781721102331629, 9.865481211320487548996411438022

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