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Properties

Label 2-1148-41.9-c1-0-4
Degree $2$
Conductor $1148$
Sign $0.241 - 0.970i$
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  + (0.254 + 0.254i)3-s + 1.56i·5-s + (0.707 + 0.707i)7-s − 2.87i·9-s + (4.48 + 4.48i)11-s + (−0.972 − 0.972i)13-s + (−0.398 + 0.398i)15-s + (−4.04 + 4.04i)17-s + (−1.81 + 1.81i)19-s + 0.360i·21-s + 2.75·23-s + 2.55·25-s + (1.49 − 1.49i)27-s + (−1.30 − 1.30i)29-s + 4.14·31-s + ⋯
L(s)  = 1  + (0.147 + 0.147i)3-s + 0.699i·5-s + (0.267 + 0.267i)7-s − 0.956i·9-s + (1.35 + 1.35i)11-s + (−0.269 − 0.269i)13-s + (−0.102 + 0.102i)15-s + (−0.981 + 0.981i)17-s + (−0.417 + 0.417i)19-s + 0.0786i·21-s + 0.574·23-s + 0.511·25-s + (0.287 − 0.287i)27-s + (−0.241 − 0.241i)29-s + 0.745·31-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(1.731313675\)
\(L(\frac12)\) \(\approx\) \(1.731313675\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.29 - 6.27i)T \)
good3 \( 1 + (-0.254 - 0.254i)T + 3iT^{2} \)
5 \( 1 - 1.56iT - 5T^{2} \)
11 \( 1 + (-4.48 - 4.48i)T + 11iT^{2} \)
13 \( 1 + (0.972 + 0.972i)T + 13iT^{2} \)
17 \( 1 + (4.04 - 4.04i)T - 17iT^{2} \)
19 \( 1 + (1.81 - 1.81i)T - 19iT^{2} \)
23 \( 1 - 2.75T + 23T^{2} \)
29 \( 1 + (1.30 + 1.30i)T + 29iT^{2} \)
31 \( 1 - 4.14T + 31T^{2} \)
37 \( 1 + 5.41T + 37T^{2} \)
43 \( 1 + 4.26iT - 43T^{2} \)
47 \( 1 + (-0.0142 + 0.0142i)T - 47iT^{2} \)
53 \( 1 + (-10.1 - 10.1i)T + 53iT^{2} \)
59 \( 1 + 14.7T + 59T^{2} \)
61 \( 1 - 13.4iT - 61T^{2} \)
67 \( 1 + (-0.159 + 0.159i)T - 67iT^{2} \)
71 \( 1 + (-9.37 - 9.37i)T + 71iT^{2} \)
73 \( 1 + 2.11iT - 73T^{2} \)
79 \( 1 + (-3.39 - 3.39i)T + 79iT^{2} \)
83 \( 1 + 9.91T + 83T^{2} \)
89 \( 1 + (-3.39 - 3.39i)T + 89iT^{2} \)
97 \( 1 + (-11.5 + 11.5i)T - 97iT^{2} \)
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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

−9.973418959694394354103157549850, −9.122516013256638789295782159206, −8.558745624738178262049707628510, −7.27489731751906328741334003549, −6.69333428807230410795128325652, −5.98813577318832937350234068150, −4.54535680104717852703894237530, −3.92229618571758875401738590464, −2.73037822096760157134226774809, −1.53006793083448939420984758385, 0.795611038689083225615759402612, 2.09340100952058631037492345257, 3.39549584878393586695501054436, 4.58066386953833619351331279919, 5.13429816052330805492426936755, 6.40454277771661398776169990017, 7.09007126576419731021812837965, 8.165258567220940755980084767369, 8.867008253619338761606727431972, 9.285410691015238148625839547727

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