Properties

Label 2-430-43.38-c1-0-13
Degree $2$
Conductor $430$
Sign $-0.594 + 0.803i$
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  + (−0.222 + 0.974i)2-s + (0.269 + 0.0830i)3-s + (−0.900 − 0.433i)4-s + (−0.365 − 0.930i)5-s + (−0.140 + 0.243i)6-s + (−0.703 − 1.21i)7-s + (0.623 − 0.781i)8-s + (−2.41 − 1.64i)9-s + (0.988 − 0.149i)10-s + (−3.05 + 1.47i)11-s + (−0.206 − 0.191i)12-s + (−5.47 − 0.825i)13-s + (1.34 − 0.414i)14-s + (−0.0210 − 0.280i)15-s + (0.623 + 0.781i)16-s + (−2.08 + 5.30i)17-s + ⋯
L(s)  = 1  + (−0.157 + 0.689i)2-s + (0.155 + 0.0479i)3-s + (−0.450 − 0.216i)4-s + (−0.163 − 0.416i)5-s + (−0.0575 + 0.0995i)6-s + (−0.265 − 0.460i)7-s + (0.220 − 0.276i)8-s + (−0.804 − 0.548i)9-s + (0.312 − 0.0471i)10-s + (−0.920 + 0.443i)11-s + (−0.0596 − 0.0553i)12-s + (−1.51 − 0.228i)13-s + (0.359 − 0.110i)14-s + (−0.00543 − 0.0725i)15-s + (0.155 + 0.195i)16-s + (−0.504 + 1.28i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.594 + 0.803i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.594 + 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.594 + 0.803i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ -0.594 + 0.803i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.101234 - 0.200836i\)
\(L(\frac12)\) \(\approx\) \(0.101234 - 0.200836i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.222 - 0.974i)T \)
5 \( 1 + (0.365 + 0.930i)T \)
43 \( 1 + (-6.17 + 2.21i)T \)
good3 \( 1 + (-0.269 - 0.0830i)T + (2.47 + 1.68i)T^{2} \)
7 \( 1 + (0.703 + 1.21i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.05 - 1.47i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (5.47 + 0.825i)T + (12.4 + 3.83i)T^{2} \)
17 \( 1 + (2.08 - 5.30i)T + (-12.4 - 11.5i)T^{2} \)
19 \( 1 + (2.81 - 1.92i)T + (6.94 - 17.6i)T^{2} \)
23 \( 1 + (-0.530 + 7.08i)T + (-22.7 - 3.42i)T^{2} \)
29 \( 1 + (-8.07 + 2.49i)T + (23.9 - 16.3i)T^{2} \)
31 \( 1 + (-0.0828 - 0.0768i)T + (2.31 + 30.9i)T^{2} \)
37 \( 1 + (3.53 - 6.12i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.772 + 3.38i)T + (-36.9 - 17.7i)T^{2} \)
47 \( 1 + (10.2 + 4.93i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (-7.24 + 1.09i)T + (50.6 - 15.6i)T^{2} \)
59 \( 1 + (-3.61 - 4.52i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (-6.08 + 5.64i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (10.0 - 6.83i)T + (24.4 - 62.3i)T^{2} \)
71 \( 1 + (-0.393 - 5.25i)T + (-70.2 + 10.5i)T^{2} \)
73 \( 1 + (5.80 + 0.875i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (6.81 + 11.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.14 + 2.20i)T + (68.5 + 46.7i)T^{2} \)
89 \( 1 + (-8.17 - 2.52i)T + (73.5 + 50.1i)T^{2} \)
97 \( 1 + (6.59 - 3.17i)T + (60.4 - 75.8i)T^{2} \)
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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

−10.37452413029363344326418895121, −10.09312024963650800882283697587, −8.675490632103735252944740535786, −8.258862099186736801753768698606, −7.12240774238358580271887965089, −6.22522225463909561281358254020, −5.06298908471398521377747526045, −4.13270579629565977602131919331, −2.53735773670888192482852580836, −0.13517238538560179362920639024, 2.48791321047149757348948822159, 2.92891306658093550606270196209, 4.69539782510691566654048934506, 5.54902652813016341576824692611, 7.04825598863421114154810422222, 7.903991229559899827003599268069, 8.923524481118809455078381271770, 9.698001917759791725324131549078, 10.70518896387742248536658748262, 11.41138425771407761248043541703

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