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Properties

Label 2-1148-41.16-c1-0-10
Degree $2$
Conductor $1148$
Sign $0.696 + 0.717i$
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  + 0.0210·3-s + (0.567 + 0.412i)5-s + (−0.309 − 0.951i)7-s − 2.99·9-s + (−0.883 + 0.641i)11-s + (0.0516 − 0.158i)13-s + (0.0119 + 0.00866i)15-s + (6.53 − 4.74i)17-s + (0.882 + 2.71i)19-s + (−0.00649 − 0.0199i)21-s + (2.42 − 7.46i)23-s + (−1.39 − 4.28i)25-s − 0.126·27-s + (6.32 + 4.59i)29-s + (2.27 − 1.65i)31-s + ⋯
L(s)  = 1  + 0.0121·3-s + (0.253 + 0.184i)5-s + (−0.116 − 0.359i)7-s − 0.999·9-s + (−0.266 + 0.193i)11-s + (0.0143 − 0.0440i)13-s + (0.00307 + 0.00223i)15-s + (1.58 − 1.15i)17-s + (0.202 + 0.622i)19-s + (−0.00141 − 0.00435i)21-s + (0.506 − 1.55i)23-s + (−0.278 − 0.857i)25-s − 0.0242·27-s + (1.17 + 0.852i)29-s + (0.408 − 0.296i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.696 + 0.717i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.696 + 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.696 + 0.717i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ 0.696 + 0.717i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.509973727\)
\(L(\frac12)\) \(\approx\) \(1.509973727\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.44 + 3.37i)T \)
good3 \( 1 - 0.0210T + 3T^{2} \)
5 \( 1 + (-0.567 - 0.412i)T + (1.54 + 4.75i)T^{2} \)
11 \( 1 + (0.883 - 0.641i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-0.0516 + 0.158i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-6.53 + 4.74i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.882 - 2.71i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-2.42 + 7.46i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-6.32 - 4.59i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.27 + 1.65i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.30 - 0.945i)T + (11.4 + 35.1i)T^{2} \)
43 \( 1 + (-1.79 + 5.53i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (-0.776 + 2.38i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (7.52 + 5.46i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (4.17 - 12.8i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.82 + 5.60i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-3.24 - 2.35i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (5.46 - 3.97i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 - 11.6T + 73T^{2} \)
79 \( 1 - 3.45T + 79T^{2} \)
83 \( 1 + 4.51T + 83T^{2} \)
89 \( 1 + (3.96 + 12.2i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (0.661 + 0.480i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.873848698573414067105160919141, −8.822537413228216183506276564063, −8.084818769842606209337536124795, −7.24129940149072619469235101230, −6.29695434190739072921891088225, −5.48225895507134081144830636945, −4.58316894727307540461230391555, −3.25350541805218755852603848693, −2.52263684576012764283163347574, −0.74731458249350066302563502530, 1.26015217928248744077650337846, 2.75241917634440811639733050740, 3.52360135981265158565406950367, 4.94922126045059232280901742069, 5.72135826237035640190261845851, 6.30564147708027065050757491407, 7.70884679933268531013286574263, 8.149751964613513259142155391183, 9.235278770182104095709971731553, 9.694367693271256791051014351585

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