Properties

Label 2-43-43.6-c3-0-7
Degree $2$
Conductor $43$
Sign $0.964 + 0.263i$
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  + 3.59·2-s + (0.838 − 1.45i)3-s + 4.91·4-s + (4.86 − 8.41i)5-s + (3.01 − 5.22i)6-s + (7.88 + 13.6i)7-s − 11.0·8-s + (12.0 + 20.9i)9-s + (17.4 − 30.2i)10-s − 59.3·11-s + (4.12 − 7.14i)12-s + (−6.63 − 11.4i)13-s + (28.3 + 49.0i)14-s + (−8.15 − 14.1i)15-s − 79.1·16-s + (14.5 + 25.1i)17-s + ⋯
L(s)  = 1  + 1.27·2-s + (0.161 − 0.279i)3-s + 0.614·4-s + (0.434 − 0.753i)5-s + (0.205 − 0.355i)6-s + (0.425 + 0.736i)7-s − 0.490·8-s + (0.447 + 0.775i)9-s + (0.552 − 0.956i)10-s − 1.62·11-s + (0.0991 − 0.171i)12-s + (−0.141 − 0.245i)13-s + (0.540 + 0.936i)14-s + (−0.140 − 0.243i)15-s − 1.23·16-s + (0.207 + 0.358i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.263i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.964 + 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.964 + 0.263i$
Analytic conductor: \(2.53708\)
Root analytic conductor: \(1.59282\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :3/2),\ 0.964 + 0.263i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.37409 - 0.317852i\)
\(L(\frac12)\) \(\approx\) \(2.37409 - 0.317852i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-250. + 129. i)T \)
good2 \( 1 - 3.59T + 8T^{2} \)
3 \( 1 + (-0.838 + 1.45i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (-4.86 + 8.41i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (-7.88 - 13.6i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + 59.3T + 1.33e3T^{2} \)
13 \( 1 + (6.63 + 11.4i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-14.5 - 25.1i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-51.8 + 89.7i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (4.27 - 7.39i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (19.7 + 34.1i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (60.4 - 104. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-164. + 284. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 274.T + 6.89e4T^{2} \)
47 \( 1 + 161.T + 1.03e5T^{2} \)
53 \( 1 + (85.6 - 148. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + 789.T + 2.05e5T^{2} \)
61 \( 1 + (260. + 451. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (222. - 385. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (428. + 741. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (-484. - 839. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-493. - 854. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (60.8 - 105. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-345. + 598. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 503.T + 9.12e5T^{2} \)
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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

−15.28762740776442375349994144539, −13.95429900866659125702986654854, −13.04303366592790820186229323781, −12.52403718381347393501741668109, −10.91694553293352657665224637048, −9.144535453222996285708660281951, −7.70413404462864558587029340220, −5.56171251709549171039945745519, −4.87543846978461909612279115104, −2.50668569690004348107101748124, 3.02591099061270886269629046107, 4.54618381919246332664311609080, 6.04362716394966428620704742402, 7.58314430023744457475563482547, 9.690881232168696354791774955906, 10.80561341976050661772736468436, 12.29074823391510328542682733326, 13.42645126911849850188976779309, 14.29974575274340033949190821840, 15.08623412070997348258441890013

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