Properties

Degree 2
Conductor 43
Sign $-0.441 + 0.897i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.98i·2-s − 41.6i·3-s + 55.0·4-s − 126. i·5-s + 124.·6-s − 69.5i·7-s + 355. i·8-s − 1.00e3·9-s + 377.·10-s − 100.·11-s − 2.29e3i·12-s − 1.72e3·13-s + 207.·14-s − 5.26e3·15-s + 2.46e3·16-s − 115.·17-s + ⋯
L(s)  = 1  + 0.373i·2-s − 1.54i·3-s + 0.860·4-s − 1.01i·5-s + 0.576·6-s − 0.202i·7-s + 0.694i·8-s − 1.38·9-s + 0.377·10-s − 0.0755·11-s − 1.32i·12-s − 0.784·13-s + 0.0757·14-s − 1.55·15-s + 0.600·16-s − 0.0235·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.441 + 0.897i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.441 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.441 + 0.897i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ -0.441 + 0.897i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(0.987282 - 1.58710i\)
\(L(\frac12)\)  \(\approx\)  \(0.987282 - 1.58710i\)
\(L(4)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 43$,\(F_p(T)\) is a polynomial of degree 2. If $p = 43$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad43 \( 1 + (-3.51e4 + 7.13e4i)T \)
good2 \( 1 - 2.98iT - 64T^{2} \)
3 \( 1 + 41.6iT - 729T^{2} \)
5 \( 1 + 126. iT - 1.56e4T^{2} \)
7 \( 1 + 69.5iT - 1.17e5T^{2} \)
11 \( 1 + 100.T + 1.77e6T^{2} \)
13 \( 1 + 1.72e3T + 4.82e6T^{2} \)
17 \( 1 + 115.T + 2.41e7T^{2} \)
19 \( 1 + 9.21e3iT - 4.70e7T^{2} \)
23 \( 1 - 6.39e3T + 1.48e8T^{2} \)
29 \( 1 - 1.54e4iT - 5.94e8T^{2} \)
31 \( 1 + 4.20e4T + 8.87e8T^{2} \)
37 \( 1 - 3.82e4iT - 2.56e9T^{2} \)
41 \( 1 - 7.86e4T + 4.75e9T^{2} \)
47 \( 1 - 1.38e5T + 1.07e10T^{2} \)
53 \( 1 - 5.48e4T + 2.21e10T^{2} \)
59 \( 1 + 1.77e5T + 4.21e10T^{2} \)
61 \( 1 + 3.72e5iT - 5.15e10T^{2} \)
67 \( 1 - 5.75e5T + 9.04e10T^{2} \)
71 \( 1 - 3.67e5iT - 1.28e11T^{2} \)
73 \( 1 - 4.75e4iT - 1.51e11T^{2} \)
79 \( 1 - 2.34e5T + 2.43e11T^{2} \)
83 \( 1 + 6.87e4T + 3.26e11T^{2} \)
89 \( 1 - 3.35e4iT - 4.96e11T^{2} \)
97 \( 1 - 3.41e5T + 8.32e11T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−14.17549312120963406695569967160, −12.91885224863799334627260205957, −12.27357534493781424544963875211, −11.05633506613531655808528082570, −8.909887583219931842098610444857, −7.58886112838320693604978238026, −6.82369676547416128864386079240, −5.30327627775963521302825880210, −2.34822331850471858502580306668, −0.879621182773044438739575679714, 2.59789374962522858603354960214, 3.86293532430408012273696120175, 5.76232027245710540873234905194, 7.39564307588810387311602488731, 9.428738587826980301286594374337, 10.45507809413341435973268911678, 11.04395356077957473366834640145, 12.34383034786804134336795989213, 14.48966760215397273560433256303, 15.04808736370064424716256854015

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