Properties

Degree 2
Conductor 43
Sign $0.292 + 0.956i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.32 + 8.97i)2-s + (20.4 − 42.4i)3-s + (−22.0 + 27.6i)4-s + (−183. − 41.9i)5-s + 469.·6-s − 28.9i·7-s + (278. + 63.5i)8-s + (−927. − 1.16e3i)9-s + (−417. − 1.83e3i)10-s + (−675. − 846. i)11-s + (721. + 1.49e3i)12-s + (485. − 2.12e3i)13-s + (260. − 125. i)14-s + (−5.53e3 + 6.93e3i)15-s + (1.13e3 + 4.98e3i)16-s + (−1.87e3 − 8.19e3i)17-s + ⋯
L(s)  = 1  + (0.540 + 1.12i)2-s + (0.756 − 1.57i)3-s + (−0.344 + 0.431i)4-s + (−1.46 − 0.335i)5-s + 2.17·6-s − 0.0845i·7-s + (0.544 + 0.124i)8-s + (−1.27 − 1.59i)9-s + (−0.417 − 1.83i)10-s + (−0.507 − 0.636i)11-s + (0.417 + 0.867i)12-s + (0.221 − 0.968i)13-s + (0.0948 − 0.0456i)14-s + (−1.63 + 2.05i)15-s + (0.277 + 1.21i)16-s + (−0.380 − 1.66i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.292 + 0.956i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (2, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ 0.292 + 0.956i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(1.68954 - 1.25041i\)
\(L(\frac12)\)  \(\approx\)  \(1.68954 - 1.25041i\)
\(L(4)\)   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 \( 1 + (-5.83e4 - 5.39e4i)T \)
good2 \( 1 + (-4.32 - 8.97i)T + (-39.9 + 50.0i)T^{2} \)
3 \( 1 + (-20.4 + 42.4i)T + (-454. - 569. i)T^{2} \)
5 \( 1 + (183. + 41.9i)T + (1.40e4 + 6.77e3i)T^{2} \)
7 \( 1 + 28.9iT - 1.17e5T^{2} \)
11 \( 1 + (675. + 846. i)T + (-3.94e5 + 1.72e6i)T^{2} \)
13 \( 1 + (-485. + 2.12e3i)T + (-4.34e6 - 2.09e6i)T^{2} \)
17 \( 1 + (1.87e3 + 8.19e3i)T + (-2.17e7 + 1.04e7i)T^{2} \)
19 \( 1 + (-6.87e3 - 5.48e3i)T + (1.04e7 + 4.58e7i)T^{2} \)
23 \( 1 + (-7.55e3 - 9.46e3i)T + (-3.29e7 + 1.44e8i)T^{2} \)
29 \( 1 + (-1.04e4 - 2.17e4i)T + (-3.70e8 + 4.65e8i)T^{2} \)
31 \( 1 + (1.95e4 - 9.42e3i)T + (5.53e8 - 6.93e8i)T^{2} \)
37 \( 1 + 1.73e4iT - 2.56e9T^{2} \)
41 \( 1 + (-9.81e3 + 4.72e3i)T + (2.96e9 - 3.71e9i)T^{2} \)
47 \( 1 + (-8.41e4 + 1.05e5i)T + (-2.39e9 - 1.05e10i)T^{2} \)
53 \( 1 + (4.05e4 + 1.77e5i)T + (-1.99e10 + 9.61e9i)T^{2} \)
59 \( 1 + (7.06e4 + 3.09e5i)T + (-3.80e10 + 1.83e10i)T^{2} \)
61 \( 1 + (2.98e4 - 6.19e4i)T + (-3.21e10 - 4.02e10i)T^{2} \)
67 \( 1 + (2.35e5 - 2.94e5i)T + (-2.01e10 - 8.81e10i)T^{2} \)
71 \( 1 + (-3.85e5 - 3.07e5i)T + (2.85e10 + 1.24e11i)T^{2} \)
73 \( 1 + (2.44e5 + 5.58e4i)T + (1.36e11 + 6.56e10i)T^{2} \)
79 \( 1 - 4.19e5T + 2.43e11T^{2} \)
83 \( 1 + (5.53e5 + 2.66e5i)T + (2.03e11 + 2.55e11i)T^{2} \)
89 \( 1 + (-5.05e4 + 1.05e5i)T + (-3.09e11 - 3.88e11i)T^{2} \)
97 \( 1 + (2.92e5 + 3.67e5i)T + (-1.85e11 + 8.12e11i)T^{2} \)
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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

−14.35748214243541688188117040459, −13.50618382096076269777721828325, −12.53879566303064353134393778737, −11.32620463930579111450978095883, −8.589164896242793170664360701762, −7.64067657777782235678988218938, −7.16617305531415110418699017219, −5.40467021292593495769821742666, −3.27823883217708273408030511597, −0.797853417668518266831513130966, 2.68417239125411542265877323615, 3.93113324978390506985725661075, 4.51534435443424675096457581653, 7.64728635940080755768800901379, 9.058853808484246649841727697460, 10.50259760413925050002550458261, 11.13002690167422943583776536934, 12.30371348444658108181653825068, 13.82443412395200452944946453242, 15.08623340139849739375200239561

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