Properties

Degree 2
Conductor 43
Sign $-0.285 - 0.958i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.43i·2-s − 6.93i·3-s − 3.69·4-s + 22.3i·5-s + 30.7·6-s + 51.9i·7-s + 54.6i·8-s + 32.8·9-s − 99.2·10-s − 25.0·11-s + 25.6i·12-s − 38.7·13-s − 230.·14-s + 155.·15-s − 301.·16-s + 111.·17-s + ⋯
L(s)  = 1  + 1.10i·2-s − 0.771i·3-s − 0.230·4-s + 0.894i·5-s + 0.855·6-s + 1.06i·7-s + 0.853i·8-s + 0.405·9-s − 0.992·10-s − 0.206·11-s + 0.177i·12-s − 0.229·13-s − 1.17·14-s + 0.689·15-s − 1.17·16-s + 0.385·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.285 - 0.958i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.285 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.285 - 0.958i$
motivic weight  =  \(4\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :2),\ -0.285 - 0.958i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.944406 + 1.26738i\)
\(L(\frac12)\)  \(\approx\)  \(0.944406 + 1.26738i\)
\(L(3)\)   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 \( 1 + (528. + 1.77e3i)T \)
good2 \( 1 - 4.43iT - 16T^{2} \)
3 \( 1 + 6.93iT - 81T^{2} \)
5 \( 1 - 22.3iT - 625T^{2} \)
7 \( 1 - 51.9iT - 2.40e3T^{2} \)
11 \( 1 + 25.0T + 1.46e4T^{2} \)
13 \( 1 + 38.7T + 2.85e4T^{2} \)
17 \( 1 - 111.T + 8.35e4T^{2} \)
19 \( 1 + 238. iT - 1.30e5T^{2} \)
23 \( 1 - 823.T + 2.79e5T^{2} \)
29 \( 1 + 424. iT - 7.07e5T^{2} \)
31 \( 1 + 1.44e3T + 9.23e5T^{2} \)
37 \( 1 + 626. iT - 1.87e6T^{2} \)
41 \( 1 - 580.T + 2.82e6T^{2} \)
47 \( 1 + 170.T + 4.87e6T^{2} \)
53 \( 1 - 4.30e3T + 7.89e6T^{2} \)
59 \( 1 + 65.4T + 1.21e7T^{2} \)
61 \( 1 - 3.89e3iT - 1.38e7T^{2} \)
67 \( 1 + 5.44e3T + 2.01e7T^{2} \)
71 \( 1 + 5.84e3iT - 2.54e7T^{2} \)
73 \( 1 - 4.77e3iT - 2.83e7T^{2} \)
79 \( 1 - 1.01e4T + 3.89e7T^{2} \)
83 \( 1 - 4.43e3T + 4.74e7T^{2} \)
89 \( 1 - 9.87e3iT - 6.27e7T^{2} \)
97 \( 1 + 7.76e3T + 8.85e7T^{2} \)
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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

−15.29112426904277435635926363677, −14.81301042167504246204230075432, −13.42592560371781921677256805500, −12.13906988356065896434497465178, −10.85777215174713415309861185511, −8.967956552969827735958371232942, −7.49174550156238721852069130842, −6.75679385434071954538918285510, −5.44178105415812301238963872529, −2.46760618385530248088167326570, 1.16354562543860040550095235321, 3.58652799255498187144706696335, 4.82798709935685033398652664996, 7.24263709797992840175899225623, 9.174039135997210294741116580549, 10.21291827274175245807107861086, 11.01282867444372784217424885705, 12.49353856907522278478572175468, 13.24035799669626406200111739996, 14.89151572871071198055180787466

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