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Properties

Label 2-1148-41.31-c1-0-10
Degree $2$
Conductor $1148$
Sign $0.557 - 0.830i$
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  + 0.268i·3-s + (0.725 + 2.23i)5-s + (0.587 + 0.809i)7-s + 2.92·9-s + (5.13 + 1.66i)11-s + (1.29 − 1.77i)13-s + (−0.598 + 0.194i)15-s + (−2.13 − 0.694i)17-s + (0.234 + 0.322i)19-s + (−0.216 + 0.157i)21-s + (−4.94 − 3.59i)23-s + (−0.411 + 0.298i)25-s + 1.58i·27-s + (−0.796 + 0.258i)29-s + (−0.226 + 0.697i)31-s + ⋯
L(s)  = 1  + 0.154i·3-s + (0.324 + 0.998i)5-s + (0.222 + 0.305i)7-s + 0.976·9-s + (1.54 + 0.502i)11-s + (0.357 − 0.492i)13-s + (−0.154 + 0.0501i)15-s + (−0.518 − 0.168i)17-s + (0.0537 + 0.0739i)19-s + (−0.0473 + 0.0343i)21-s + (−1.03 − 0.749i)23-s + (−0.0822 + 0.0597i)25-s + 0.305i·27-s + (−0.147 + 0.0480i)29-s + (−0.0406 + 0.125i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.557 - 0.830i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ 0.557 - 0.830i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.068851503\)
\(L(\frac12)\) \(\approx\) \(2.068851503\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.587 - 0.809i)T \)
41 \( 1 + (-5.06 - 3.91i)T \)
good3 \( 1 - 0.268iT - 3T^{2} \)
5 \( 1 + (-0.725 - 2.23i)T + (-4.04 + 2.93i)T^{2} \)
11 \( 1 + (-5.13 - 1.66i)T + (8.89 + 6.46i)T^{2} \)
13 \( 1 + (-1.29 + 1.77i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (2.13 + 0.694i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-0.234 - 0.322i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + (4.94 + 3.59i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (0.796 - 0.258i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.226 - 0.697i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (1.56 + 4.80i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (-6.26 - 4.54i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (5.50 - 7.57i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (0.775 - 0.251i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-0.625 - 0.454i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-4.04 + 2.93i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-3.57 + 1.16i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (-0.416 - 0.135i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + 5.48T + 73T^{2} \)
79 \( 1 - 5.94iT - 79T^{2} \)
83 \( 1 + 4.84T + 83T^{2} \)
89 \( 1 + (3.57 + 4.91i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (4.29 - 1.39i)T + (78.4 - 57.0i)T^{2} \)
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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

−9.838831339881285985046020561314, −9.355121475159369991361704166995, −8.297646585411179059119650989931, −7.27751606778127971825086466545, −6.59210617603049261357940152882, −5.94768534841165032916450835853, −4.55265740834314685442734681721, −3.86413199590375783165520847016, −2.60933172207671480773382156204, −1.47883403318429440054698382320, 1.09177916577639968552582653498, 1.85231628718190293865873433098, 3.81110358799483143944649674384, 4.28098101380991926056473572584, 5.40963908736988808416514688627, 6.38852303819725928008972908009, 7.09091439085507881539719504830, 8.156920202531214387016372114532, 8.980417482646062556610669246635, 9.469271838380703191018492834628

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