Properties

Label 2-43-43.24-c1-0-1
Degree $2$
Conductor $43$
Sign $0.325 - 0.945i$
Analytic cond. $0.343356$
Root an. cond. $0.585966$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.44 + 1.80i)2-s + (2.66 − 0.402i)3-s + (−0.745 − 3.26i)4-s + (−2.09 + 1.43i)5-s + (−3.12 + 5.40i)6-s + (−1.09 − 1.89i)7-s + (2.80 + 1.35i)8-s + (4.09 − 1.26i)9-s + (0.438 − 5.85i)10-s + (0.694 − 3.04i)11-s + (−3.30 − 8.41i)12-s + (0.257 + 3.43i)13-s + (5.00 + 0.753i)14-s + (−5.02 + 4.66i)15-s + (−0.462 + 0.222i)16-s + (−1.92 − 1.31i)17-s + ⋯
L(s)  = 1  + (−1.01 + 1.27i)2-s + (1.54 − 0.232i)3-s + (−0.372 − 1.63i)4-s + (−0.938 + 0.639i)5-s + (−1.27 + 2.20i)6-s + (−0.413 − 0.715i)7-s + (0.993 + 0.478i)8-s + (1.36 − 0.421i)9-s + (0.138 − 1.85i)10-s + (0.209 − 0.917i)11-s + (−0.953 − 2.42i)12-s + (0.0713 + 0.951i)13-s + (1.33 + 0.201i)14-s + (−1.29 + 1.20i)15-s + (−0.115 + 0.0557i)16-s + (−0.467 − 0.318i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 - 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.325 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.325 - 0.945i$
Analytic conductor: \(0.343356\)
Root analytic conductor: \(0.585966\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :1/2),\ 0.325 - 0.945i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.528855 + 0.377317i\)
\(L(\frac12)\) \(\approx\) \(0.528855 + 0.377317i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-6.03 - 2.55i)T \)
good2 \( 1 + (1.44 - 1.80i)T + (-0.445 - 1.94i)T^{2} \)
3 \( 1 + (-2.66 + 0.402i)T + (2.86 - 0.884i)T^{2} \)
5 \( 1 + (2.09 - 1.43i)T + (1.82 - 4.65i)T^{2} \)
7 \( 1 + (1.09 + 1.89i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.694 + 3.04i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (-0.257 - 3.43i)T + (-12.8 + 1.93i)T^{2} \)
17 \( 1 + (1.92 + 1.31i)T + (6.21 + 15.8i)T^{2} \)
19 \( 1 + (3.23 + 0.996i)T + (15.6 + 10.7i)T^{2} \)
23 \( 1 + (-4.82 - 4.47i)T + (1.71 + 22.9i)T^{2} \)
29 \( 1 + (5.34 + 0.806i)T + (27.7 + 8.54i)T^{2} \)
31 \( 1 + (-0.778 - 1.98i)T + (-22.7 + 21.0i)T^{2} \)
37 \( 1 + (1.73 - 3.01i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.33 + 1.66i)T + (-9.12 - 39.9i)T^{2} \)
47 \( 1 + (-0.260 - 1.14i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (-0.777 + 10.3i)T + (-52.4 - 7.89i)T^{2} \)
59 \( 1 + (5.53 - 2.66i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-0.913 + 2.32i)T + (-44.7 - 41.4i)T^{2} \)
67 \( 1 + (-9.87 - 3.04i)T + (55.3 + 37.7i)T^{2} \)
71 \( 1 + (-8.30 + 7.70i)T + (5.30 - 70.8i)T^{2} \)
73 \( 1 + (-0.624 - 8.33i)T + (-72.1 + 10.8i)T^{2} \)
79 \( 1 + (4.90 + 8.50i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.83 - 0.427i)T + (79.3 - 24.4i)T^{2} \)
89 \( 1 + (-14.8 + 2.23i)T + (85.0 - 26.2i)T^{2} \)
97 \( 1 + (-0.939 + 4.11i)T + (-87.3 - 42.0i)T^{2} \)
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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

−16.06425190635400477543536383849, −15.19849791362882648860495787915, −14.35791934489161544383032104066, −13.39282381080707402361169958396, −11.11603654464193959532172599933, −9.455729013079964191248719849030, −8.589495480557880019998233813887, −7.49259483904133812086635581207, −6.76247377004281882929780867013, −3.58174178444803421268793409390, 2.50294294458104842656796662973, 3.97870418589518055973641404751, 7.81092590874933374475136101538, 8.714469437232136101159624235638, 9.394327397059009432195327302858, 10.72529677955583619361140701085, 12.35899897272329567872899411185, 12.88043839528537690857063835280, 14.87534263706378308766205917702, 15.59238184180793197579905540557

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