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Properties

Label 2-43-43.36-c3-0-3
Degree $2$
Conductor $43$
Sign $0.747 + 0.663i$
Analytic cond. $2.53708$
Root an. cond. $1.59282$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.55·2-s + (0.194 + 0.336i)3-s + 4.66·4-s + (0.863 + 1.49i)5-s + (−0.692 − 1.19i)6-s + (10.5 − 18.3i)7-s + 11.8·8-s + (13.4 − 23.2i)9-s + (−3.07 − 5.32i)10-s + 23.7·11-s + (0.907 + 1.57i)12-s + (15.0 − 26.1i)13-s + (−37.6 + 65.2i)14-s + (−0.336 + 0.582i)15-s − 79.5·16-s + (−25.8 + 44.7i)17-s + ⋯
L(s)  = 1  − 1.25·2-s + (0.0374 + 0.0648i)3-s + 0.582·4-s + (0.0772 + 0.133i)5-s + (−0.0471 − 0.0815i)6-s + (0.571 − 0.990i)7-s + 0.524·8-s + (0.497 − 0.861i)9-s + (−0.0972 − 0.168i)10-s + 0.650·11-s + (0.0218 + 0.0377i)12-s + (0.321 − 0.557i)13-s + (−0.719 + 1.24i)14-s + (−0.00578 + 0.0100i)15-s − 1.24·16-s + (−0.368 + 0.638i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.747 + 0.663i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.747 + 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.747 + 0.663i$
Analytic conductor: \(2.53708\)
Root analytic conductor: \(1.59282\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :3/2),\ 0.747 + 0.663i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.710683 - 0.269883i\)
\(L(\frac12)\) \(\approx\) \(0.710683 - 0.269883i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-28.8 - 280. i)T \)
good2 \( 1 + 3.55T + 8T^{2} \)
3 \( 1 + (-0.194 - 0.336i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (-0.863 - 1.49i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + (-10.5 + 18.3i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 - 23.7T + 1.33e3T^{2} \)
13 \( 1 + (-15.0 + 26.1i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (25.8 - 44.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (14.6 + 25.3i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-46.7 - 80.9i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-29.8 + 51.6i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (56.7 + 98.3i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (34.3 + 59.4i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 53.9T + 6.89e4T^{2} \)
47 \( 1 + 455.T + 1.03e5T^{2} \)
53 \( 1 + (-331. - 573. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + 457.T + 2.05e5T^{2} \)
61 \( 1 + (-303. + 525. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-214. - 371. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-69.9 + 121. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (240. - 417. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (552. - 956. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-628. - 1.08e3i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (312. + 540. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.12e3T + 9.12e5T^{2} \)
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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

−15.55643355737425218273313039679, −14.29147113248272146471889112381, −13.00556937527552889757583342316, −11.26715052968961912851922177203, −10.29531363074401327624224340713, −9.196634220568065972340766456228, −7.938808649485629950194371605069, −6.69808239595492571576651065175, −4.18829055725000379583201329154, −1.09160112756577451804488613223, 1.77809969348581544267197127501, 4.88393737485035992793716603823, 7.00158448937933303199651807348, 8.435502572383798454275457250112, 9.185942208624619728345738421522, 10.59288963674809648940732604969, 11.71173674177314806120626790753, 13.25838318811264902387296204373, 14.57146948133574423290826466412, 16.01924995821020798454999837308

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