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Properties

Label 2-430-43.15-c1-0-7
Degree $2$
Conductor $430$
Sign $0.997 + 0.0634i$
Analytic cond. $3.43356$
Root an. cond. $1.85298$
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  + (−0.623 + 0.781i)2-s + (0.396 − 1.01i)3-s + (−0.222 − 0.974i)4-s + (0.0747 + 0.997i)5-s + (0.543 + 0.940i)6-s + (0.223 − 0.386i)7-s + (0.900 + 0.433i)8-s + (1.33 + 1.23i)9-s + (−0.826 − 0.563i)10-s + (1.28 − 5.61i)11-s + (−1.07 − 0.161i)12-s + (0.373 − 0.254i)13-s + (0.163 + 0.415i)14-s + (1.03 + 0.320i)15-s + (−0.900 + 0.433i)16-s + (−0.296 + 3.95i)17-s + ⋯
L(s)  = 1  + (−0.440 + 0.552i)2-s + (0.229 − 0.583i)3-s + (−0.111 − 0.487i)4-s + (0.0334 + 0.445i)5-s + (0.221 + 0.384i)6-s + (0.0844 − 0.146i)7-s + (0.318 + 0.153i)8-s + (0.444 + 0.412i)9-s + (−0.261 − 0.178i)10-s + (0.386 − 1.69i)11-s + (−0.310 − 0.0467i)12-s + (0.103 − 0.0706i)13-s + (0.0436 + 0.111i)14-s + (0.267 + 0.0826i)15-s + (−0.225 + 0.108i)16-s + (−0.0718 + 0.958i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0634i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.997 + 0.0634i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ 0.997 + 0.0634i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28601 - 0.0408547i\)
\(L(\frac12)\) \(\approx\) \(1.28601 - 0.0408547i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.623 - 0.781i)T \)
5 \( 1 + (-0.0747 - 0.997i)T \)
43 \( 1 + (5.94 + 2.76i)T \)
good3 \( 1 + (-0.396 + 1.01i)T + (-2.19 - 2.04i)T^{2} \)
7 \( 1 + (-0.223 + 0.386i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.28 + 5.61i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (-0.373 + 0.254i)T + (4.74 - 12.1i)T^{2} \)
17 \( 1 + (0.296 - 3.95i)T + (-16.8 - 2.53i)T^{2} \)
19 \( 1 + (0.327 - 0.304i)T + (1.41 - 18.9i)T^{2} \)
23 \( 1 + (-7.04 + 2.17i)T + (19.0 - 12.9i)T^{2} \)
29 \( 1 + (2.44 + 6.23i)T + (-21.2 + 19.7i)T^{2} \)
31 \( 1 + (-7.48 - 1.12i)T + (29.6 + 9.13i)T^{2} \)
37 \( 1 + (-2.40 - 4.16i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-6.05 + 7.59i)T + (-9.12 - 39.9i)T^{2} \)
47 \( 1 + (-0.856 - 3.75i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (-0.0159 - 0.0108i)T + (19.3 + 49.3i)T^{2} \)
59 \( 1 + (7.61 - 3.66i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (3.03 - 0.458i)T + (58.2 - 17.9i)T^{2} \)
67 \( 1 + (9.95 - 9.23i)T + (5.00 - 66.8i)T^{2} \)
71 \( 1 + (9.82 + 3.02i)T + (58.6 + 39.9i)T^{2} \)
73 \( 1 + (8.00 - 5.46i)T + (26.6 - 67.9i)T^{2} \)
79 \( 1 + (-5.58 + 9.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.964 - 2.45i)T + (-60.8 - 56.4i)T^{2} \)
89 \( 1 + (-3.60 + 9.17i)T + (-65.2 - 60.5i)T^{2} \)
97 \( 1 + (4.02 - 17.6i)T + (-87.3 - 42.0i)T^{2} \)
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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

−10.88140256304743230357578709528, −10.39296216668858392225911346445, −9.049970952947217578770808306848, −8.308146713613233298994772897254, −7.53424137000709069183949158580, −6.52548552517656789809914213469, −5.84408652732865830416248201538, −4.32539287604753500388424624946, −2.83968077834291676853385664131, −1.16663580247940684270205370862, 1.43356221019151723967962474003, 2.97420762096002755248457056054, 4.31849183341507807600435290952, 4.95750760278852053220889295400, 6.75327276892599361288203252386, 7.54529376568916292759331827312, 8.896912792480604536345615056060, 9.427122136924523402319722190373, 10.01640146783798222497510180350, 11.09840760516939624309256113043

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