Properties

Label 2-43-43.14-c1-0-2
Degree $2$
Conductor $43$
Sign $0.937 + 0.349i$
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  + (0.993 − 0.478i)2-s + (−0.0482 − 0.644i)3-s + (−0.488 + 0.612i)4-s + (−3.17 + 0.979i)5-s + (−0.356 − 0.616i)6-s + (1.23 − 2.13i)7-s + (−0.683 + 2.99i)8-s + (2.55 − 0.384i)9-s + (−2.68 + 2.49i)10-s + (0.748 + 0.937i)11-s + (0.418 + 0.285i)12-s + (−4.22 − 3.91i)13-s + (0.203 − 2.71i)14-s + (0.784 + 1.99i)15-s + (0.404 + 1.77i)16-s + (1.02 + 0.314i)17-s + ⋯
L(s)  = 1  + (0.702 − 0.338i)2-s + (−0.0278 − 0.371i)3-s + (−0.244 + 0.306i)4-s + (−1.41 + 0.437i)5-s + (−0.145 − 0.251i)6-s + (0.465 − 0.807i)7-s + (−0.241 + 1.05i)8-s + (0.851 − 0.128i)9-s + (−0.849 + 0.788i)10-s + (0.225 + 0.282i)11-s + (0.120 + 0.0823i)12-s + (−1.17 − 1.08i)13-s + (0.0542 − 0.724i)14-s + (0.202 + 0.515i)15-s + (0.101 + 0.442i)16-s + (0.247 + 0.0763i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 + 0.349i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.937 + 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.937 + 0.349i$
Analytic conductor: \(0.343356\)
Root analytic conductor: \(0.585966\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :1/2),\ 0.937 + 0.349i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.882537 - 0.159099i\)
\(L(\frac12)\) \(\approx\) \(0.882537 - 0.159099i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (2.45 - 6.08i)T \)
good2 \( 1 + (-0.993 + 0.478i)T + (1.24 - 1.56i)T^{2} \)
3 \( 1 + (0.0482 + 0.644i)T + (-2.96 + 0.447i)T^{2} \)
5 \( 1 + (3.17 - 0.979i)T + (4.13 - 2.81i)T^{2} \)
7 \( 1 + (-1.23 + 2.13i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.748 - 0.937i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (4.22 + 3.91i)T + (0.971 + 12.9i)T^{2} \)
17 \( 1 + (-1.02 - 0.314i)T + (14.0 + 9.57i)T^{2} \)
19 \( 1 + (-2.13 - 0.321i)T + (18.1 + 5.60i)T^{2} \)
23 \( 1 + (2.16 - 5.50i)T + (-16.8 - 15.6i)T^{2} \)
29 \( 1 + (-0.177 + 2.37i)T + (-28.6 - 4.32i)T^{2} \)
31 \( 1 + (-6.31 - 4.30i)T + (11.3 + 28.8i)T^{2} \)
37 \( 1 + (2.83 + 4.90i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (8.77 - 4.22i)T + (25.5 - 32.0i)T^{2} \)
47 \( 1 + (-3.30 + 4.14i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-2.69 + 2.50i)T + (3.96 - 52.8i)T^{2} \)
59 \( 1 + (-0.208 - 0.914i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (2.31 - 1.57i)T + (22.2 - 56.7i)T^{2} \)
67 \( 1 + (-11.8 - 1.78i)T + (64.0 + 19.7i)T^{2} \)
71 \( 1 + (5.54 + 14.1i)T + (-52.0 + 48.2i)T^{2} \)
73 \( 1 + (4.99 + 4.63i)T + (5.45 + 72.7i)T^{2} \)
79 \( 1 + (1.18 - 2.05i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.533 + 7.11i)T + (-82.0 + 12.3i)T^{2} \)
89 \( 1 + (0.113 + 1.51i)T + (-88.0 + 13.2i)T^{2} \)
97 \( 1 + (6.73 + 8.44i)T + (-21.5 + 94.5i)T^{2} \)
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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

−15.63390677859709095263942171346, −14.69400915501883701229240746908, −13.50107169342892283080034547592, −12.30624625935540003129003199618, −11.67200370311089545013438426045, −10.19018922086412880174489790580, −7.935630213240924471869940932621, −7.34492575041471470739952230894, −4.72765157710429876239442766895, −3.50638318597277095651229242111, 4.14847543939525586155832743892, 5.02116596424919735914380159025, 7.01691791649351277509195021052, 8.598481195985574487263055552796, 9.949245027890213825257009428564, 11.75892552234353425131369422470, 12.42321461776912538179371393555, 14.02444045653120162285715066052, 15.10630382801623908956626801524, 15.70525618997617741537877993065

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