Properties

Label 2-43-43.6-c5-0-7
Degree $2$
Conductor $43$
Sign $-0.126 - 0.991i$
Analytic cond. $6.89650$
Root an. cond. $2.62611$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.36·2-s + (0.424 − 0.735i)3-s − 12.9·4-s + (−30.5 + 52.8i)5-s + (1.85 − 3.21i)6-s + (93.7 + 162. i)7-s − 196.·8-s + (121. + 209. i)9-s + (−133. + 230. i)10-s + 227.·11-s + (−5.49 + 9.52i)12-s + (−266. − 461. i)13-s + (409. + 708. i)14-s + (25.9 + 44.9i)15-s − 442.·16-s + (−497. − 862. i)17-s + ⋯
L(s)  = 1  + 0.771·2-s + (0.0272 − 0.0472i)3-s − 0.404·4-s + (−0.546 + 0.946i)5-s + (0.0210 − 0.0364i)6-s + (0.722 + 1.25i)7-s − 1.08·8-s + (0.498 + 0.863i)9-s + (−0.421 + 0.730i)10-s + 0.567·11-s + (−0.0110 + 0.0190i)12-s + (−0.437 − 0.757i)13-s + (0.557 + 0.966i)14-s + (0.0297 + 0.0515i)15-s − 0.431·16-s + (−0.417 − 0.723i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.126 - 0.991i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.126 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.126 - 0.991i$
Analytic conductor: \(6.89650\)
Root analytic conductor: \(2.62611\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :5/2),\ -0.126 - 0.991i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.14431 + 1.29935i\)
\(L(\frac12)\) \(\approx\) \(1.14431 + 1.29935i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-9.25e3 - 7.82e3i)T \)
good2 \( 1 - 4.36T + 32T^{2} \)
3 \( 1 + (-0.424 + 0.735i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (30.5 - 52.8i)T + (-1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (-93.7 - 162. i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 - 227.T + 1.61e5T^{2} \)
13 \( 1 + (266. + 461. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + (497. + 862. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (1.04e3 - 1.80e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (-574. + 994. i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (-2.12e3 - 3.68e3i)T + (-1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (-369. + 640. i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-8.28e3 + 1.43e4i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 - 8.48e3T + 1.15e8T^{2} \)
47 \( 1 - 8.27e3T + 2.29e8T^{2} \)
53 \( 1 + (1.40e4 - 2.43e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 - 8.72e3T + 7.14e8T^{2} \)
61 \( 1 + (-2.07e4 - 3.59e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (2.10e4 - 3.65e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (7.14e3 + 1.23e4i)T + (-9.02e8 + 1.56e9i)T^{2} \)
73 \( 1 + (3.97e4 + 6.87e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (2.53e4 + 4.39e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (-3.63e4 + 6.28e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + (-2.28e4 + 3.95e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + 1.04e5T + 8.58e9T^{2} \)
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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

−14.79311039429299488845538903283, −14.44806274110838621851979994752, −12.88071554094855644251606327809, −11.92840212502485979633672990893, −10.69656980296535430023153795043, −8.975856576516300921623866924169, −7.61279726227992021432285892779, −5.80443288626557619079366234184, −4.44371344733715937826598773202, −2.65771831480667071451190016326, 0.794457863064576057927519898763, 4.11788797611165263798491228219, 4.55625445576868699920285325584, 6.70264336815725118601605178885, 8.402949163966807508476106553794, 9.565767919740217465277904377814, 11.38772051524214200297407869731, 12.47562506039789548506733130146, 13.43822831005731057042269495754, 14.50780696997224215632771950227

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