Properties

Label 2-1148-7.2-c1-0-18
Degree $2$
Conductor $1148$
Sign $0.890 + 0.455i$
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  + (0.966 + 1.67i)3-s + (0.237 − 0.411i)5-s + (−1.29 − 2.30i)7-s + (−0.366 + 0.635i)9-s + (−0.304 − 0.526i)11-s + 0.258·13-s + 0.918·15-s + (−3.33 − 5.78i)17-s + (4.27 − 7.41i)19-s + (2.60 − 4.39i)21-s + (0.270 − 0.468i)23-s + (2.38 + 4.13i)25-s + 4.37·27-s + 9.17·29-s + (1.59 + 2.76i)31-s + ⋯
L(s)  = 1  + (0.557 + 0.966i)3-s + (0.106 − 0.184i)5-s + (−0.489 − 0.872i)7-s + (−0.122 + 0.211i)9-s + (−0.0917 − 0.158i)11-s + 0.0716·13-s + 0.237·15-s + (−0.809 − 1.40i)17-s + (0.981 − 1.70i)19-s + (0.569 − 0.959i)21-s + (0.0563 − 0.0976i)23-s + (0.477 + 0.826i)25-s + 0.842·27-s + 1.70·29-s + (0.286 + 0.496i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 + 0.455i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.890 + 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.890 + 0.455i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (821, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ 0.890 + 0.455i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.828165051\)
\(L(\frac12)\) \(\approx\) \(1.828165051\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.29 + 2.30i)T \)
41 \( 1 + T \)
good3 \( 1 + (-0.966 - 1.67i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.237 + 0.411i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.304 + 0.526i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.258T + 13T^{2} \)
17 \( 1 + (3.33 + 5.78i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.27 + 7.41i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.270 + 0.468i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 9.17T + 29T^{2} \)
31 \( 1 + (-1.59 - 2.76i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.05 - 1.83i)T + (-18.5 - 32.0i)T^{2} \)
43 \( 1 + 9.45T + 43T^{2} \)
47 \( 1 + (-2.60 + 4.51i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.56 + 9.64i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.40 - 5.88i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.701 - 1.21i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.99 + 13.8i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.72T + 71T^{2} \)
73 \( 1 + (-4.65 - 8.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.73 - 4.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 0.971T + 83T^{2} \)
89 \( 1 + (4.89 - 8.47i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.26T + 97T^{2} \)
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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

−9.646867000358534555242438423361, −9.123190146847967069266533579530, −8.361407023792300364423142213853, −7.03478848579399441834394853351, −6.72148656517559860707060780892, −4.99322957007772889539251131491, −4.67517625541276590937377505148, −3.40361081014556271691855079593, −2.81985278143931295089387060522, −0.792228578457759797687666372002, 1.49839120894028340523580778155, 2.43841483684307743484532091127, 3.37919394612829046327601400819, 4.68966751106888387233965411352, 6.01273665836108569814254567846, 6.42228578533163625819820768430, 7.47590461321172926083049313898, 8.301565620664653666896373952873, 8.724673105622642405062296189419, 9.935966654275357251335835993483

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