Properties

Label 2-43-43.22-c6-0-6
Degree $2$
Conductor $43$
Sign $0.292 - 0.956i$
Analytic cond. $9.89232$
Root an. cond. $3.14520$
Motivic weight $6$
Arithmetic yes
Rational no
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

Degree: \(2\)
Conductor: \(43\)
Sign: $0.292 - 0.956i$
Analytic conductor: \(9.89232\)
Root analytic conductor: \(3.14520\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :3),\ 0.292 - 0.956i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.68954 + 1.25041i\)
\(L(\frac12)\) \(\approx\) \(1.68954 + 1.25041i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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