Properties

Degree 2
Conductor 43
Sign $-0.246 + 0.969i$
Motivic weight 9
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.74 + 4.70i)2-s + (29.2 + 4.40i)3-s + (105. − 463. i)4-s + (328. + 224. i)5-s + (88.8 + 153. i)6-s + (2.91e3 − 5.05e3i)7-s + (5.35e3 − 2.57e3i)8-s + (−1.79e4 − 5.54e3i)9-s + (178. + 2.38e3i)10-s + (1.14e4 + 5.03e4i)11-s + (5.13e3 − 1.30e4i)12-s + (1.02e4 − 1.36e5i)13-s + (3.47e4 − 5.23e3i)14-s + (8.62e3 + 8.00e3i)15-s + (−1.87e5 − 9.01e4i)16-s + (−2.45e5 + 1.67e5i)17-s + ⋯
L(s)  = 1  + (0.165 + 0.207i)2-s + (0.208 + 0.0314i)3-s + (0.206 − 0.906i)4-s + (0.235 + 0.160i)5-s + (0.0280 + 0.0484i)6-s + (0.459 − 0.795i)7-s + (0.462 − 0.222i)8-s + (−0.913 − 0.281i)9-s + (0.00565 + 0.0755i)10-s + (0.236 + 1.03i)11-s + (0.0715 − 0.182i)12-s + (0.0994 − 1.32i)13-s + (0.241 − 0.0363i)14-s + (0.0439 + 0.0408i)15-s + (−0.714 − 0.344i)16-s + (−0.713 + 0.486i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.246 + 0.969i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.246 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.246 + 0.969i$
motivic weight  =  \(9\)
character  :  $\chi_{43} (9, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :9/2),\ -0.246 + 0.969i)\)
\(L(5)\)  \(\approx\)  \(1.20084 - 1.54380i\)
\(L(\frac12)\)  \(\approx\)  \(1.20084 - 1.54380i\)
\(L(\frac{11}{2})\)   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 \( 1 + (-2.20e7 - 4.00e6i)T \)
good2 \( 1 + (-3.74 - 4.70i)T + (-113. + 499. i)T^{2} \)
3 \( 1 + (-29.2 - 4.40i)T + (1.88e4 + 5.80e3i)T^{2} \)
5 \( 1 + (-328. - 224. i)T + (7.13e5 + 1.81e6i)T^{2} \)
7 \( 1 + (-2.91e3 + 5.05e3i)T + (-2.01e7 - 3.49e7i)T^{2} \)
11 \( 1 + (-1.14e4 - 5.03e4i)T + (-2.12e9 + 1.02e9i)T^{2} \)
13 \( 1 + (-1.02e4 + 1.36e5i)T + (-1.04e10 - 1.58e9i)T^{2} \)
17 \( 1 + (2.45e5 - 1.67e5i)T + (4.33e10 - 1.10e11i)T^{2} \)
19 \( 1 + (4.76e5 - 1.46e5i)T + (2.66e11 - 1.81e11i)T^{2} \)
23 \( 1 + (-1.37e6 + 1.27e6i)T + (1.34e11 - 1.79e12i)T^{2} \)
29 \( 1 + (-2.86e6 + 4.31e5i)T + (1.38e13 - 4.27e12i)T^{2} \)
31 \( 1 + (-3.97e5 + 1.01e6i)T + (-1.93e13 - 1.79e13i)T^{2} \)
37 \( 1 + (1.11e7 + 1.93e7i)T + (-6.49e13 + 1.12e14i)T^{2} \)
41 \( 1 + (1.69e7 + 2.13e7i)T + (-7.28e13 + 3.19e14i)T^{2} \)
47 \( 1 + (3.75e6 - 1.64e7i)T + (-1.00e15 - 4.85e14i)T^{2} \)
53 \( 1 + (-2.71e6 - 3.62e7i)T + (-3.26e15 + 4.91e14i)T^{2} \)
59 \( 1 + (-7.64e7 - 3.68e7i)T + (5.40e15 + 6.77e15i)T^{2} \)
61 \( 1 + (-3.73e7 - 9.52e7i)T + (-8.57e15 + 7.95e15i)T^{2} \)
67 \( 1 + (7.60e7 - 2.34e7i)T + (2.24e16 - 1.53e16i)T^{2} \)
71 \( 1 + (-2.92e8 - 2.71e8i)T + (3.42e15 + 4.57e16i)T^{2} \)
73 \( 1 + (-1.63e7 + 2.18e8i)T + (-5.82e16 - 8.77e15i)T^{2} \)
79 \( 1 + (7.89e7 - 1.36e8i)T + (-5.99e16 - 1.03e17i)T^{2} \)
83 \( 1 + (7.08e8 + 1.06e8i)T + (1.78e17 + 5.51e16i)T^{2} \)
89 \( 1 + (-1.60e8 - 2.41e7i)T + (3.34e17 + 1.03e17i)T^{2} \)
97 \( 1 + (1.03e8 + 4.55e8i)T + (-6.84e17 + 3.29e17i)T^{2} \)
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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

−13.99109595588792600666759619206, −12.60300385308581853497568272564, −10.88257633694423882916219111983, −10.24395690667937790559036116879, −8.642761631641955412227397216707, −7.07607551476233800456235031260, −5.81962019702173384889600874644, −4.35937929725734882649989704550, −2.27417334342150615272587429688, −0.59979419332916720407687548523, 1.96886232656644482959545439578, 3.24172425953246228143038963443, 4.99351248939961959532788756901, 6.66807140351432082302092690527, 8.408666862843259860484540800501, 8.975611862609500114918296075318, 11.31897770906897671517796350804, 11.64384062634824024653441993064, 13.25795755308487011976779278298, 14.03513822595000207209764741979

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