Properties

Label 2-43-43.15-c1-0-2
Degree $2$
Conductor $43$
Sign $0.997 - 0.0712i$
Analytic cond. $0.343356$
Root an. cond. $0.585966$
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  + (−0.651 + 0.816i)2-s + (0.922 − 2.34i)3-s + (0.202 + 0.885i)4-s + (−0.0373 − 0.498i)5-s + (1.31 + 2.28i)6-s + (−1.65 + 2.86i)7-s + (−2.73 − 1.31i)8-s + (−2.47 − 2.29i)9-s + (0.431 + 0.294i)10-s + (0.828 − 3.62i)11-s + (2.26 + 0.341i)12-s + (−3.87 + 2.64i)13-s + (−1.26 − 3.21i)14-s + (−1.20 − 0.371i)15-s + (1.22 − 0.589i)16-s + (−0.343 + 4.57i)17-s + ⋯
L(s)  = 1  + (−0.460 + 0.577i)2-s + (0.532 − 1.35i)3-s + (0.101 + 0.442i)4-s + (−0.0167 − 0.222i)5-s + (0.538 + 0.932i)6-s + (−0.624 + 1.08i)7-s + (−0.967 − 0.466i)8-s + (−0.823 − 0.764i)9-s + (0.136 + 0.0930i)10-s + (0.249 − 1.09i)11-s + (0.654 + 0.0986i)12-s + (−1.07 + 0.732i)13-s + (−0.337 − 0.859i)14-s + (−0.311 − 0.0960i)15-s + (0.305 − 0.147i)16-s + (−0.0831 + 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.997 - 0.0712i$
Analytic conductor: \(0.343356\)
Root analytic conductor: \(0.585966\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :1/2),\ 0.997 - 0.0712i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.709139 + 0.0252997i\)
\(L(\frac12)\) \(\approx\) \(0.709139 + 0.0252997i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-6.26 + 1.92i)T \)
good2 \( 1 + (0.651 - 0.816i)T + (-0.445 - 1.94i)T^{2} \)
3 \( 1 + (-0.922 + 2.34i)T + (-2.19 - 2.04i)T^{2} \)
5 \( 1 + (0.0373 + 0.498i)T + (-4.94 + 0.745i)T^{2} \)
7 \( 1 + (1.65 - 2.86i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.828 + 3.62i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (3.87 - 2.64i)T + (4.74 - 12.1i)T^{2} \)
17 \( 1 + (0.343 - 4.57i)T + (-16.8 - 2.53i)T^{2} \)
19 \( 1 + (-4.44 + 4.12i)T + (1.41 - 18.9i)T^{2} \)
23 \( 1 + (0.371 - 0.114i)T + (19.0 - 12.9i)T^{2} \)
29 \( 1 + (1.29 + 3.29i)T + (-21.2 + 19.7i)T^{2} \)
31 \( 1 + (-0.861 - 0.129i)T + (29.6 + 9.13i)T^{2} \)
37 \( 1 + (1.96 + 3.40i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.72 - 8.43i)T + (-9.12 - 39.9i)T^{2} \)
47 \( 1 + (0.776 + 3.40i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (-3.94 - 2.68i)T + (19.3 + 49.3i)T^{2} \)
59 \( 1 + (6.05 - 2.91i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (5.18 - 0.782i)T + (58.2 - 17.9i)T^{2} \)
67 \( 1 + (7.73 - 7.17i)T + (5.00 - 66.8i)T^{2} \)
71 \( 1 + (-0.859 - 0.265i)T + (58.6 + 39.9i)T^{2} \)
73 \( 1 + (0.798 - 0.544i)T + (26.6 - 67.9i)T^{2} \)
79 \( 1 + (-0.500 + 0.867i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.24 - 3.17i)T + (-60.8 - 56.4i)T^{2} \)
89 \( 1 + (0.672 - 1.71i)T + (-65.2 - 60.5i)T^{2} \)
97 \( 1 + (0.213 - 0.936i)T + (-87.3 - 42.0i)T^{2} \)
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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

−16.18299620135581693796233233453, −14.95872205216282782626460012821, −13.56309044234343189139956223878, −12.53693245791985938283728951860, −11.80559141485463960891385417015, −9.210205040509821794413450110109, −8.454509187819405584679614690320, −7.20792796097456308017256313059, −6.14276547953100547990607515674, −2.81110447078383814848124626173, 3.17966145664464763119425229737, 4.94582305626491947305856745732, 7.22974631711017457486682902709, 9.309328748172286981697204593091, 9.991557700351727751689529686289, 10.58860523752104873676597569150, 12.20045451861484728016500051899, 14.08205403372406528374435551423, 14.88972951595294269873900116371, 15.86445550026294444604884465290

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