Properties

Label 2-43-43.9-c1-0-2
Degree $2$
Conductor $43$
Sign $-0.851 + 0.525i$
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  + (−1.11 − 1.39i)2-s + (−2.93 − 0.442i)3-s + (−0.263 + 1.15i)4-s + (−0.492 − 0.335i)5-s + (2.64 + 4.58i)6-s + (2.18 − 3.77i)7-s + (−1.31 + 0.632i)8-s + (5.54 + 1.71i)9-s + (0.0795 + 1.06i)10-s + (−0.452 − 1.98i)11-s + (1.28 − 3.26i)12-s + (−0.130 + 1.73i)13-s + (−7.69 + 1.16i)14-s + (1.29 + 1.20i)15-s + (4.47 + 2.15i)16-s + (1.21 − 0.826i)17-s + ⋯
L(s)  = 1  + (−0.786 − 0.986i)2-s + (−1.69 − 0.255i)3-s + (−0.131 + 0.576i)4-s + (−0.220 − 0.150i)5-s + (1.08 + 1.87i)6-s + (0.824 − 1.42i)7-s + (−0.464 + 0.223i)8-s + (1.84 + 0.570i)9-s + (0.0251 + 0.335i)10-s + (−0.136 − 0.598i)11-s + (0.370 − 0.943i)12-s + (−0.0361 + 0.482i)13-s + (−2.05 + 0.310i)14-s + (0.334 + 0.310i)15-s + (1.11 + 0.538i)16-s + (0.294 − 0.200i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.851 + 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.851 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.851 + 0.525i$
Analytic conductor: \(0.343356\)
Root analytic conductor: \(0.585966\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :1/2),\ -0.851 + 0.525i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0909643 - 0.320666i\)
\(L(\frac12)\) \(\approx\) \(0.0909643 - 0.320666i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-3.81 - 5.32i)T \)
good2 \( 1 + (1.11 + 1.39i)T + (-0.445 + 1.94i)T^{2} \)
3 \( 1 + (2.93 + 0.442i)T + (2.86 + 0.884i)T^{2} \)
5 \( 1 + (0.492 + 0.335i)T + (1.82 + 4.65i)T^{2} \)
7 \( 1 + (-2.18 + 3.77i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.452 + 1.98i)T + (-9.91 + 4.77i)T^{2} \)
13 \( 1 + (0.130 - 1.73i)T + (-12.8 - 1.93i)T^{2} \)
17 \( 1 + (-1.21 + 0.826i)T + (6.21 - 15.8i)T^{2} \)
19 \( 1 + (2.07 - 0.641i)T + (15.6 - 10.7i)T^{2} \)
23 \( 1 + (-1.99 + 1.85i)T + (1.71 - 22.9i)T^{2} \)
29 \( 1 + (-7.48 + 1.12i)T + (27.7 - 8.54i)T^{2} \)
31 \( 1 + (0.208 - 0.530i)T + (-22.7 - 21.0i)T^{2} \)
37 \( 1 + (-1.98 - 3.43i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.32 - 1.66i)T + (-9.12 + 39.9i)T^{2} \)
47 \( 1 + (2.08 - 9.12i)T + (-42.3 - 20.3i)T^{2} \)
53 \( 1 + (0.441 + 5.88i)T + (-52.4 + 7.89i)T^{2} \)
59 \( 1 + (4.60 + 2.21i)T + (36.7 + 46.1i)T^{2} \)
61 \( 1 + (3.57 + 9.10i)T + (-44.7 + 41.4i)T^{2} \)
67 \( 1 + (0.506 - 0.156i)T + (55.3 - 37.7i)T^{2} \)
71 \( 1 + (-3.91 - 3.63i)T + (5.30 + 70.8i)T^{2} \)
73 \( 1 + (1.05 - 14.0i)T + (-72.1 - 10.8i)T^{2} \)
79 \( 1 + (1.82 - 3.16i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-10.2 - 1.54i)T + (79.3 + 24.4i)T^{2} \)
89 \( 1 + (-5.97 - 0.899i)T + (85.0 + 26.2i)T^{2} \)
97 \( 1 + (2.28 + 9.99i)T + (-87.3 + 42.0i)T^{2} \)
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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

−16.16063107877768933584853157012, −14.17283677906146033729535455068, −12.59385088946712694388547816894, −11.49402481838221974567725188215, −10.92614207298278185854770008052, −10.05856813467542783834335183183, −8.039642874087825213679877145370, −6.40166147079653704767739797978, −4.61456906422842171681142969854, −0.951633784206193250715239967742, 5.13566128304468705190497729781, 6.10959513661904506147022534468, 7.52818269411036852534605274152, 8.995118290924563013297866809266, 10.47371021517231411855563051179, 11.77283409742061445358469792704, 12.46081300441712166786907905455, 15.08873291305536092760529770707, 15.48504439234753408853042429297, 16.62378239829328268021307521520

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